### Example 1: Using the factorization method solve the quadratic equation: x^{2} +24x +143= 0

Let's rewrite this equation as: x^{2} +24x +143= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +143x^{2} while the sum remains +24x

x^{2 }+ 13x + 11x +143= 0

=> x(x + 13) + 11 (x + 13) = 0

=> (x + 13) (x + 11) = 0

=> (x + 13) =0 OR (x + 11) = 0

=> x = - 13 OR x = - 11

**=> So, by factorization, we solve and find that the roots of the given equation are - 13 , - 11**

### Example 2: Using the factorization method solve the quadratic equation: 2x^{2} -x -231= 0

Let's rewrite this equation as: 2x^{2} -x -231= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -462x^{2} (product of the first and the last term) while the sum remains -x

2x^{2 }+ 21x - 22x -231= 0

=> x(2x + 21) - 11 (2x + 21) = 0

=> (2x + 21) (x - 22) = 0

=> (2x + 21) =0 OR (x - 11) = 0

=> x = - 21/2 OR x = + 11

**=> So, by factorization, we solve and find that the roots of the given equation are - 21/2 , + 11**

### Example 3: Using the factorization method solve the quadratic equation: 15x^{2} -161x +300= 0

Let's rewrite this equation as: 15x^{2} -161x +300= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +300x^{2} while the sum remains -161x

15 x^{2 }- 36x - 125x +300= 0

=> 3x(5x - 12) - 25 (5x - 12) = 0

=> (5x - 12) (3x - 25) = 0

=> (5x - 12) =0 OR (3x - 25) = 0

=> x = + 12/5 OR x = + 25/3

**=> So, by factorization, we solve and find that the roots of the given equation are + 12/5 , + 25/3**

### Example 4: Using the factorization method solve the quadratic equation: x^{2} +10x -119= 0

Let's rewrite this equation as: x^{2} +10x -119= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -119x^{2} while the sum remains +10x

x^{2 }+ 17x - 7x -119= 0

=> x(x + 17) - 7 (x + 17) = 0

=> (x + 17) (x - 7) = 0

=> (x + 17) =0 OR (x - 7) = 0

=> x = - 17 OR x = + 7

**=> So, by factorization, we solve and find that the roots of the given equation are - 17 , + 7**

### Example 5: Using the factorization method solve the quadratic equation: x^{2} -9x -136= 0

Let's rewrite this equation as: x^{2} -9x -136= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -136x^{2} while the sum remains -9x

x^{2 }- 17x + 8x -136= 0

=> x(x - 17) + 8 (x - 17) = 0

=> (x - 17) (x + 8) = 0

=> (x - 17) =0 OR (x + 8) = 0

=> x = + 17 OR x = - 8

**=> So, by factorization, we solve and find that the roots of the given equation are + 17 , - 8**

### Example 6: Using the factorization method solve the quadratic equation: x^{2} +3x +2= 0

Let's rewrite this equation as: x^{2} +3x +2= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +2x^{2} while the sum remains +3x

x^{2 }+ 1x + 2x +2= 0

=> x(x + 1) + 2 (x + 1) = 0

=> (x + 1) (x + 2) = 0

=> (x + 1) =0 OR (x + 2) = 0

=> x = - 1 OR x = - 2

**=> So, by factorization, we solve and find that the roots of the given equation are - 1 , - 2**

### Example 7: Using the factorization method solve the quadratic equation: x^{2} -10x -39= 0

Let's rewrite this equation as: x^{2} -10x -39= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -39x^{2} while the sum remains -10x

x^{2 }- 13x + 3x -39= 0

=> x(x - 13) + 3 (x - 13) = 0

=> (x - 13) (x + 3) = 0

=> (x - 13) =0 OR (x + 3) = 0

=> x = + 13 OR x = - 3

**=> So, by factorization, we solve and find that the roots of the given equation are + 13 , - 3**

### Example 8: Using the factorization method solve the quadratic equation: x^{2} +23x +102= 0

Let's rewrite this equation as: x^{2} +23x +102= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +102x^{2} while the sum remains +23x

x^{2 }+ 6x + 17x +102= 0

=> x(x + 6) + 17 (x + 6) = 0

=> (x + 6) (x + 17) = 0

=> (x + 6) =0 OR (x + 17) = 0

=> x = - 6 OR x = - 17

**=> So, by factorization, we solve and find that the roots of the given equation are - 6 , - 17**

### Example 9: Using the factorization method solve the quadratic equation: x^{2} -16x +39= 0

Let's rewrite this equation as: x^{2} -16x +39= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +39x^{2} while the sum remains -16x

x^{2 }- 3x - 13x +39= 0

=> x(x - 3) - 13 (x - 3) = 0

=> (x - 3) (x - 13) = 0

=> (x - 3) =0 OR (x - 13) = 0

=> x = + 3 OR x = + 13

**=> So, by factorization, we solve and find that the roots of the given equation are + 3 , + 13**

### Example 10: Using the factorization method solve the quadratic equation: x^{2} -19x +70= 0

Let's rewrite this equation as: x^{2} -19x +70= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +70x^{2} while the sum remains -19x

x^{2 }- 14x - 5x +70= 0

=> x(x - 14) - 5 (x - 14) = 0

=> (x - 14) (x - 5) = 0

=> (x - 14) =0 OR (x - 5) = 0

=> x = + 14 OR x = + 5

**=> So, by factorization, we solve and find that the roots of the given equation are + 14 , + 5**

### Example 11: Using the factorization method solve the quadratic equation: x^{2} -18x +17= 0

Let's rewrite this equation as: x^{2} -18x +17= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +17x^{2} while the sum remains -18x

x^{2 }- 17x - 1x +17= 0

=> x(x - 17) - 1 (x - 17) = 0

=> (x - 17) (x - 1) = 0

=> (x - 17) =0 OR (x - 1) = 0

=> x = + 17 OR x = + 1

**=> So, by factorization, we solve and find that the roots of the given equation are + 17 , + 1**

### Example 12: Using the factorization method solve the quadratic equation: x^{2} +26x +88= 0

Let's rewrite this equation as: x^{2} +26x +88= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +88x^{2} while the sum remains +26x

x^{2 }+ 4x + 22x +88= 0

=> x(x + 4) + 22 (x + 4) = 0

=> (x + 4) (x + 22) = 0

=> (x + 4) =0 OR (x + 22) = 0

=> x = - 4 OR x = - 22

**=> So, by factorization, we solve and find that the roots of the given equation are - 4 , - 22**

### Example 13: Using the factorization method solve the quadratic equation: x^{2} +16x +15= 0

Let's rewrite this equation as: x^{2} +16x +15= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +15x^{2} while the sum remains +16x

x^{2 }+ 15x + 1x +15= 0

=> x(x + 15) + 1 (x + 15) = 0

=> (x + 15) (x + 1) = 0

=> (x + 15) =0 OR (x + 1) = 0

=> x = - 15 OR x = - 1

**=> So, by factorization, we solve and find that the roots of the given equation are - 15 , - 1**

### Example 14: Using the factorization method solve the quadratic equation: x^{2} +25x +46= 0

Let's rewrite this equation as: x^{2} +25x +46= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +46x^{2} while the sum remains +25x

x^{2 }+ 23x + 2x +46= 0

=> x(x + 23) + 2 (x + 23) = 0

=> (x + 23) (x + 2) = 0

=> (x + 23) =0 OR (x + 2) = 0

=> x = - 23 OR x = - 2

**=> So, by factorization, we solve and find that the roots of the given equation are - 23 , - 2**

### Example 15: Using the factorization method solve the quadratic equation: x^{2} -17x +72= 0

Let's rewrite this equation as: x^{2} -17x +72= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +72x^{2} while the sum remains -17x

x^{2 }- 9x - 8x +72= 0

=> x(x - 9) - 8 (x - 9) = 0

=> (x - 9) (x - 8) = 0

=> (x - 9) =0 OR (x - 8) = 0

=> x = + 9 OR x = + 8

**=> So, by factorization, we solve and find that the roots of the given equation are + 9 , + 8**

### Example 16: Using the factorization method solve the quadratic equation: x^{2} -4x -117= 0

Let's rewrite this equation as: x^{2} -4x -117= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -117x^{2} while the sum remains -4x

x^{2 }- 13x + 9x -117= 0

=> x(x - 13) + 9 (x - 13) = 0

=> (x - 13) (x + 9) = 0

=> (x - 13) =0 OR (x + 9) = 0

=> x = + 13 OR x = - 9

**=> So, by factorization, we solve and find that the roots of the given equation are + 13 , - 9**

### Example 17: Using the factorization method solve the quadratic equation: x^{2} +23x +22= 0

Let's rewrite this equation as: x^{2} +23x +22= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +22x^{2} while the sum remains +23x

x^{2 }+ 1x + 22x +22= 0

=> x(x + 1) + 22 (x + 1) = 0

=> (x + 1) (x + 22) = 0

=> (x + 1) =0 OR (x + 22) = 0

=> x = - 1 OR x = - 22

**=> So, by factorization, we solve and find that the roots of the given equation are - 1 , - 22**

### Example 18: Using the factorization method solve the quadratic equation: x^{2} -10x -56= 0

Let's rewrite this equation as: x^{2} -10x -56= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -56x^{2} while the sum remains -10x

x^{2 }- 14x + 4x -56= 0

=> x(x - 14) + 4 (x - 14) = 0

=> (x - 14) (x + 4) = 0

=> (x - 14) =0 OR (x + 4) = 0

=> x = + 14 OR x = - 4

**=> So, by factorization, we solve and find that the roots of the given equation are + 14 , - 4**

### Example 19: Using the factorization method solve the quadratic equation: x^{2} -33x +266= 0

Let's rewrite this equation as: x^{2} -33x +266= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +266x^{2} while the sum remains -33x

x^{2 }- 14x - 19x +266= 0

=> x(x - 14) - 19 (x - 14) = 0

=> (x - 14) (x - 19) = 0

=> (x - 14) =0 OR (x - 19) = 0

=> x = + 14 OR x = + 19

**=> So, by factorization, we solve and find that the roots of the given equation are + 14 , + 19**

### Example 20: Using the factorization method solve the quadratic equation: x^{2} +26x +133= 0

Let's rewrite this equation as: x^{2} +26x +133= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +133x^{2} while the sum remains +26x

x^{2 }+ 7x + 19x +133= 0

=> x(x + 7) + 19 (x + 7) = 0

=> (x + 7) (x + 19) = 0

=> (x + 7) =0 OR (x + 19) = 0

=> x = - 7 OR x = - 19

**=> So, by factorization, we solve and find that the roots of the given equation are - 7 , - 19**

### Example 21: Using the factorization method solve the quadratic equation: x^{2} +0x -169= 0

Let's rewrite this equation as: x^{2} +0x -169= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -169x^{2} while the sum remains +0x

x^{2 }- 13x + 13x -169= 0

=> x(x - 13) + 13 (x - 13) = 0

=> (x - 13) (x + 13) = 0

=> (x - 13) =0 OR (x + 13) = 0

=> x = + 13 OR x = - 13

**=> So, by factorization, we solve and find that the roots of the given equation are + 13 , - 13**

### Example 22: Using the factorization method solve the quadratic equation: x^{2} -11x -350= 0

Let's rewrite this equation as: x^{2} -11x -350= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -350x^{2} while the sum remains -11x

x^{2 }- 25x + 14x -350= 0

=> x(x - 25) + 14 (x - 25) = 0

=> (x - 25) (x + 14) = 0

=> (x - 25) =0 OR (x + 14) = 0

=> x = + 25 OR x = - 14

**=> So, by factorization, we solve and find that the roots of the given equation are + 25 , - 14**

### Example 23: Using the factorization method solve the quadratic equation: x^{2} +25x +126= 0

Let's rewrite this equation as: x^{2} +25x +126= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +126x^{2} while the sum remains +25x

x^{2 }+ 7x + 18x +126= 0

=> x(x + 7) + 18 (x + 7) = 0

=> (x + 7) (x + 18) = 0

=> (x + 7) =0 OR (x + 18) = 0

=> x = - 7 OR x = - 18

**=> So, by factorization, we solve and find that the roots of the given equation are - 7 , - 18**

### Example 24: Using the factorization method solve the quadratic equation: x^{2} +12x +11= 0

Let's rewrite this equation as: x^{2} +12x +11= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +11x^{2} while the sum remains +12x

x^{2 }+ 1x + 11x +11= 0

=> x(x + 1) + 11 (x + 1) = 0

=> (x + 1) (x + 11) = 0

=> (x + 1) =0 OR (x + 11) = 0

=> x = - 1 OR x = - 11

**=> So, by factorization, we solve and find that the roots of the given equation are - 1 , - 11**

### Example 25: Using the factorization method solve the quadratic equation: x^{2} +45x +506= 0

Let's rewrite this equation as: x^{2} +45x +506= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +506x^{2} while the sum remains +45x

x^{2 }+ 22x + 23x +506= 0

=> x(x + 22) + 23 (x + 22) = 0

=> (x + 22) (x + 23) = 0

=> (x + 22) =0 OR (x + 23) = 0

=> x = - 22 OR x = - 23

**=> So, by factorization, we solve and find that the roots of the given equation are - 22 , - 23**

### Example 26: Using the factorization method solve the quadratic equation: x^{2} +0x -121= 0

Let's rewrite this equation as: x^{2} +0x -121= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -121x^{2} while the sum remains +0x

x^{2 }- 11x + 11x -121= 0

=> x(x - 11) + 11 (x - 11) = 0

=> (x - 11) (x + 11) = 0

=> (x - 11) =0 OR (x + 11) = 0

=> x = + 11 OR x = - 11

**=> So, by factorization, we solve and find that the roots of the given equation are + 11 , - 11**

### Example 27: Using the factorization method solve the quadratic equation: x^{2} +22x +57= 0

Let's rewrite this equation as: x^{2} +22x +57= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +57x^{2} while the sum remains +22x

x^{2 }+ 19x + 3x +57= 0

=> x(x + 19) + 3 (x + 19) = 0

=> (x + 19) (x + 3) = 0

=> (x + 19) =0 OR (x + 3) = 0

=> x = - 19 OR x = - 3

**=> So, by factorization, we solve and find that the roots of the given equation are - 19 , - 3**

### Example 28: Using the factorization method solve the quadratic equation: x^{2} -30x +144= 0

Let's rewrite this equation as: x^{2} -30x +144= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +144x^{2} while the sum remains -30x

x^{2 }- 24x - 6x +144= 0

=> x(x - 24) - 6 (x - 24) = 0

=> (x - 24) (x - 6) = 0

=> (x - 24) =0 OR (x - 6) = 0

=> x = + 24 OR x = + 6

**=> So, by factorization, we solve and find that the roots of the given equation are + 24 , + 6**

### Example 29: Using the factorization method solve the quadratic equation: x^{2} -26x +48= 0

Let's rewrite this equation as: x^{2} -26x +48= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +48x^{2} while the sum remains -26x

x^{2 }- 2x - 24x +48= 0

=> x(x - 2) - 24 (x - 2) = 0

=> (x - 2) (x - 24) = 0

=> (x - 2) =0 OR (x - 24) = 0

=> x = + 2 OR x = + 24

**=> So, by factorization, we solve and find that the roots of the given equation are + 2 , + 24**

### Example 30: Using the factorization method solve the quadratic equation: x^{2} -28x +195= 0

Let's rewrite this equation as: x^{2} -28x +195= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +195x^{2} while the sum remains -28x

x^{2 }- 15x - 13x +195= 0

=> x(x - 15) - 13 (x - 15) = 0

=> (x - 15) (x - 13) = 0

=> (x - 15) =0 OR (x - 13) = 0

=> x = + 15 OR x = + 13

**=> So, by factorization, we solve and find that the roots of the given equation are + 15 , + 13**

### Example 31: Using the factorization method solve the quadratic equation: x^{2} +15x +50= 0

Let's rewrite this equation as: x^{2} +15x +50= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +50x^{2} while the sum remains +15x

x^{2 }+ 10x + 5x +50= 0

=> x(x + 10) + 5 (x + 10) = 0

=> (x + 10) (x + 5) = 0

=> (x + 10) =0 OR (x + 5) = 0

=> x = - 10 OR x = - 5

**=> So, by factorization, we solve and find that the roots of the given equation are - 10 , - 5**

### Example 32: Using the factorization method solve the quadratic equation: x^{2} -5x -374= 0

Let's rewrite this equation as: x^{2} -5x -374= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -374x^{2} while the sum remains -5x

x^{2 }+ 17x - 22x -374= 0

=> x(x + 17) - 22 (x + 17) = 0

=> (x + 17) (x - 22) = 0

=> (x + 17) =0 OR (x - 22) = 0

=> x = - 17 OR x = + 22

**=> So, by factorization, we solve and find that the roots of the given equation are - 17 , + 22**

### Example 33: Using the factorization method solve the quadratic equation: x^{2} -4x -60= 0

Let's rewrite this equation as: x^{2} -4x -60= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -60x^{2} while the sum remains -4x

x^{2 }+ 6x - 10x -60= 0

=> x(x + 6) - 10 (x + 6) = 0

=> (x + 6) (x - 10) = 0

=> (x + 6) =0 OR (x - 10) = 0

=> x = - 6 OR x = + 10

**=> So, by factorization, we solve and find that the roots of the given equation are - 6 , + 10**

### Example 34: Using the factorization method solve the quadratic equation: x^{2} +12x +35= 0

Let's rewrite this equation as: x^{2} +12x +35= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +35x^{2} while the sum remains +12x

x^{2 }+ 5x + 7x +35= 0

=> x(x + 5) + 7 (x + 5) = 0

=> (x + 5) (x + 7) = 0

=> (x + 5) =0 OR (x + 7) = 0

=> x = - 5 OR x = - 7

**=> So, by factorization, we solve and find that the roots of the given equation are - 5 , - 7**

### Example 35: Using the factorization method solve the quadratic equation: x^{2} +27x +170= 0

Let's rewrite this equation as: x^{2} +27x +170= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +170x^{2} while the sum remains +27x

x^{2 }+ 10x + 17x +170= 0

=> x(x + 10) + 17 (x + 10) = 0

=> (x + 10) (x + 17) = 0

=> (x + 10) =0 OR (x + 17) = 0

=> x = - 10 OR x = - 17

**=> So, by factorization, we solve and find that the roots of the given equation are - 10 , - 17**

### Example 36: Using the factorization method solve the quadratic equation: x^{2} -9x -360= 0

Let's rewrite this equation as: x^{2} -9x -360= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -360x^{2} while the sum remains -9x

x^{2 }- 24x + 15x -360= 0

=> x(x - 24) + 15 (x - 24) = 0

=> (x - 24) (x + 15) = 0

=> (x - 24) =0 OR (x + 15) = 0

=> x = + 24 OR x = - 15

**=> So, by factorization, we solve and find that the roots of the given equation are + 24 , - 15**

### Example 37: Using the factorization method solve the quadratic equation: x^{2} +17x +60= 0

Let's rewrite this equation as: x^{2} +17x +60= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +60x^{2} while the sum remains +17x

x^{2 }+ 5x + 12x +60= 0

=> x(x + 5) + 12 (x + 5) = 0

=> (x + 5) (x + 12) = 0

=> (x + 5) =0 OR (x + 12) = 0

=> x = - 5 OR x = - 12

**=> So, by factorization, we solve and find that the roots of the given equation are - 5 , - 12**

### Example 38: Using the factorization method solve the quadratic equation: x^{2} +13x -198= 0

Let's rewrite this equation as: x^{2} +13x -198= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -198x^{2} while the sum remains +13x

x^{2 }+ 22x - 9x -198= 0

=> x(x + 22) - 9 (x + 22) = 0

=> (x + 22) (x - 9) = 0

=> (x + 22) =0 OR (x - 9) = 0

=> x = - 22 OR x = + 9

**=> So, by factorization, we solve and find that the roots of the given equation are - 22 , + 9**

### Example 39: Using the factorization method solve the quadratic equation: x^{2} +0x -289= 0

Let's rewrite this equation as: x^{2} +0x -289= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -289x^{2} while the sum remains +0x

x^{2 }- 17x + 17x -289= 0

=> x(x - 17) + 17 (x - 17) = 0

=> (x - 17) (x + 17) = 0

=> (x - 17) =0 OR (x + 17) = 0

=> x = + 17 OR x = - 17

**=> So, by factorization, we solve and find that the roots of the given equation are + 17 , - 17**

### Example 40: Using the factorization method solve the quadratic equation: x^{2} +24x +128= 0

Let's rewrite this equation as: x^{2} +24x +128= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +128x^{2} while the sum remains +24x

x^{2 }+ 8x + 16x +128= 0

=> x(x + 8) + 16 (x + 8) = 0

=> (x + 8) (x + 16) = 0

=> (x + 8) =0 OR (x + 16) = 0

=> x = - 8 OR x = - 16

**=> So, by factorization, we solve and find that the roots of the given equation are - 8 , - 16**

### Example 41: Using the factorization method solve the quadratic equation: x^{2} +12x +27= 0

Let's rewrite this equation as: x^{2} +12x +27= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +27x^{2} while the sum remains +12x

x^{2 }+ 9x + 3x +27= 0

=> x(x + 9) + 3 (x + 9) = 0

=> (x + 9) (x + 3) = 0

=> (x + 9) =0 OR (x + 3) = 0

=> x = - 9 OR x = - 3

**=> So, by factorization, we solve and find that the roots of the given equation are - 9 , - 3**

### Example 42: Using the factorization method solve the quadratic equation: x^{2} +18x +77= 0

Let's rewrite this equation as: x^{2} +18x +77= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +77x^{2} while the sum remains +18x

x^{2 }+ 7x + 11x +77= 0

=> x(x + 7) + 11 (x + 7) = 0

=> (x + 7) (x + 11) = 0

=> (x + 7) =0 OR (x + 11) = 0

=> x = - 7 OR x = - 11

**=> So, by factorization, we solve and find that the roots of the given equation are - 7 , - 11**

### Example 43: Using the factorization method solve the quadratic equation: x^{2} -9x -10= 0

Let's rewrite this equation as: x^{2} -9x -10= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -10x^{2} while the sum remains -9x

x^{2 }+ 1x - 10x -10= 0

=> x(x + 1) - 10 (x + 1) = 0

=> (x + 1) (x - 10) = 0

=> (x + 1) =0 OR (x - 10) = 0

=> x = - 1 OR x = + 10

**=> So, by factorization, we solve and find that the roots of the given equation are - 1 , + 10**

### Example 44: Using the factorization method solve the quadratic equation: x^{2} +43x +456= 0

Let's rewrite this equation as: x^{2} +43x +456= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +456x^{2} while the sum remains +43x

x^{2 }+ 24x + 19x +456= 0

=> x(x + 24) + 19 (x + 24) = 0

=> (x + 24) (x + 19) = 0

=> (x + 24) =0 OR (x + 19) = 0

=> x = - 24 OR x = - 19

**=> So, by factorization, we solve and find that the roots of the given equation are - 24 , - 19**

### Example 45: Using the factorization method solve the quadratic equation: x^{2} +13x +12= 0

Let's rewrite this equation as: x^{2} +13x +12= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +12x^{2} while the sum remains +13x

x^{2 }+ 12x + 1x +12= 0

=> x(x + 12) + 1 (x + 12) = 0

=> (x + 12) (x + 1) = 0

=> (x + 12) =0 OR (x + 1) = 0

=> x = - 12 OR x = - 1

**=> So, by factorization, we solve and find that the roots of the given equation are - 12 , - 1**

### Example 46: Using the factorization method solve the quadratic equation: x^{2} -11x -12= 0

Let's rewrite this equation as: x^{2} -11x -12= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -12x^{2} while the sum remains -11x

x^{2 }- 12x + 1x -12= 0

=> x(x - 12) + 1 (x - 12) = 0

=> (x - 12) (x + 1) = 0

=> (x - 12) =0 OR (x + 1) = 0

=> x = + 12 OR x = - 1

**=> So, by factorization, we solve and find that the roots of the given equation are + 12 , - 1**

### Example 47: Using the factorization method solve the quadratic equation: x^{2} +8x -153= 0

Let's rewrite this equation as: x^{2} +8x -153= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -153x^{2} while the sum remains +8x

x^{2 }+ 17x - 9x -153= 0

=> x(x + 17) - 9 (x + 17) = 0

=> (x + 17) (x - 9) = 0

=> (x + 17) =0 OR (x - 9) = 0

=> x = - 17 OR x = + 9

**=> So, by factorization, we solve and find that the roots of the given equation are - 17 , + 9**

### Example 48: Using the factorization method solve the quadratic equation: x^{2} +26x +144= 0

Let's rewrite this equation as: x^{2} +26x +144= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +144x^{2} while the sum remains +26x

x^{2 }+ 8x + 18x +144= 0

=> x(x + 8) + 18 (x + 8) = 0

=> (x + 8) (x + 18) = 0

=> (x + 8) =0 OR (x + 18) = 0

=> x = - 8 OR x = - 18

**=> So, by factorization, we solve and find that the roots of the given equation are - 8 , - 18**

### Example 49: Using the factorization method solve the quadratic equation: x^{2} +24x +119= 0

Let's rewrite this equation as: x^{2} +24x +119= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +119x^{2} while the sum remains +24x

x^{2 }+ 7x + 17x +119= 0

=> x(x + 7) + 17 (x + 7) = 0

=> (x + 7) (x + 17) = 0

=> (x + 7) =0 OR (x + 17) = 0

=> x = - 7 OR x = - 17

**=> So, by factorization, we solve and find that the roots of the given equation are - 7 , - 17**

### Example 50: Using the factorization method solve the quadratic equation: x^{2} +8x -9= 0

Let's rewrite this equation as: x^{2} +8x -9= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -9x^{2} while the sum remains +8x

x^{2 }- 1x + 9x -9= 0

=> x(x - 1) + 9 (x - 1) = 0

=> (x - 1) (x + 9) = 0

=> (x - 1) =0 OR (x + 9) = 0

=> x = + 1 OR x = - 9

**=> So, by factorization, we solve and find that the roots of the given equation are + 1 , - 9**

### Example 51: Using the factorization method solve the quadratic equation: x^{2} +40x +384= 0

Let's rewrite this equation as: x^{2} +40x +384= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +384x^{2} while the sum remains +40x

x^{2 }+ 24x + 16x +384= 0

=> x(x + 24) + 16 (x + 24) = 0

=> (x + 24) (x + 16) = 0

=> (x + 24) =0 OR (x + 16) = 0

=> x = - 24 OR x = - 16

**=> So, by factorization, we solve and find that the roots of the given equation are - 24 , - 16**

### Example 52: Using the factorization method solve the quadratic equation: x^{2} +4x -252= 0

Let's rewrite this equation as: x^{2} +4x -252= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -252x^{2} while the sum remains +4x

x^{2 }- 14x + 18x -252= 0

=> x(x - 14) + 18 (x - 14) = 0

=> (x - 14) (x + 18) = 0

=> (x - 14) =0 OR (x + 18) = 0

=> x = + 14 OR x = - 18

**=> So, by factorization, we solve and find that the roots of the given equation are + 14 , - 18**

### Example 53: Using the factorization method solve the quadratic equation: x^{2} +10x +9= 0

Let's rewrite this equation as: x^{2} +10x +9= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +9x^{2} while the sum remains +10x

x^{2 }+ 9x + 1x +9= 0

=> x(x + 9) + 1 (x + 9) = 0

=> (x + 9) (x + 1) = 0

=> (x + 9) =0 OR (x + 1) = 0

=> x = - 9 OR x = - 1

**=> So, by factorization, we solve and find that the roots of the given equation are - 9 , - 1**

### Example 54: Using the factorization method solve the quadratic equation: x^{2} +x -506= 0

Let's rewrite this equation as: x^{2} +x -506= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -506x^{2} while the sum remains +x

x^{2 }- 22x + 23x -506= 0

=> x(x - 22) + 23 (x - 22) = 0

=> (x - 22) (x + 23) = 0

=> (x - 22) =0 OR (x + 23) = 0

=> x = + 22 OR x = - 23

**=> So, by factorization, we solve and find that the roots of the given equation are + 22 , - 23**

### Example 55: Using the factorization method solve the quadratic equation: x^{2} +14x +49= 0

Let's rewrite this equation as: x^{2} +14x +49= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +49x^{2} while the sum remains +14x

x^{2 }+ 7x + 7x +49= 0

=> x(x + 7) + 7 (x + 7) = 0

=> (x + 7) (x + 7) = 0

=> (x + 7) =0 OR (x + 7) = 0

=> x = - 7 OR x = - 7

**=> So, by factorization, we solve and find that the roots of the given equation are - 7 , - 7**

### Example 56: Using the factorization method solve the quadratic equation: x^{2} -18x -144= 0

Let's rewrite this equation as: x^{2} -18x -144= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -144x^{2} while the sum remains -18x

x^{2 }- 24x + 6x -144= 0

=> x(x - 24) + 6 (x - 24) = 0

=> (x - 24) (x + 6) = 0

=> (x - 24) =0 OR (x + 6) = 0

=> x = + 24 OR x = - 6

**=> So, by factorization, we solve and find that the roots of the given equation are + 24 , - 6**

### Example 57: Using the factorization method solve the quadratic equation: x^{2} -5x -50= 0

Let's rewrite this equation as: x^{2} -5x -50= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -50x^{2} while the sum remains -5x

x^{2 }+ 5x - 10x -50= 0

=> x(x + 5) - 10 (x + 5) = 0

=> (x + 5) (x - 10) = 0

=> (x + 5) =0 OR (x - 10) = 0

=> x = - 5 OR x = + 10

**=> So, by factorization, we solve and find that the roots of the given equation are - 5 , + 10**

### Example 58: Using the factorization method solve the quadratic equation: x^{2} +5x -84= 0

Let's rewrite this equation as: x^{2} +5x -84= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -84x^{2} while the sum remains +5x

x^{2 }- 7x + 12x -84= 0

=> x(x - 7) + 12 (x - 7) = 0

=> (x - 7) (x + 12) = 0

=> (x - 7) =0 OR (x + 12) = 0

=> x = + 7 OR x = - 12

**=> So, by factorization, we solve and find that the roots of the given equation are + 7 , - 12**

### Example 59: Using the factorization method solve the quadratic equation: x^{2} -21x -22= 0

Let's rewrite this equation as: x^{2} -21x -22= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -22x^{2} while the sum remains -21x

x^{2 }- 22x + 1x -22= 0

=> x(x - 22) + 1 (x - 22) = 0

=> (x - 22) (x + 1) = 0

=> (x - 22) =0 OR (x + 1) = 0

=> x = + 22 OR x = - 1

**=> So, by factorization, we solve and find that the roots of the given equation are + 22 , - 1**

### Example 60: Using the factorization method solve the quadratic equation: x^{2} +36x +308= 0

Let's rewrite this equation as: x^{2} +36x +308= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +308x^{2} while the sum remains +36x

x^{2 }+ 22x + 14x +308= 0

=> x(x + 22) + 14 (x + 22) = 0

=> (x + 22) (x + 14) = 0

=> (x + 22) =0 OR (x + 14) = 0

=> x = - 22 OR x = - 14

**=> So, by factorization, we solve and find that the roots of the given equation are - 22 , - 14**

### Example 61: Using the factorization method solve the quadratic equation: x^{2} +25x +24= 0

Let's rewrite this equation as: x^{2} +25x +24= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +24x^{2} while the sum remains +25x

x^{2 }+ 24x + 1x +24= 0

=> x(x + 24) + 1 (x + 24) = 0

=> (x + 24) (x + 1) = 0

=> (x + 24) =0 OR (x + 1) = 0

=> x = - 24 OR x = - 1

**=> So, by factorization, we solve and find that the roots of the given equation are - 24 , - 1**

### Example 62: Using the factorization method solve the quadratic equation: x^{2} -4x -221= 0

Let's rewrite this equation as: x^{2} -4x -221= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -221x^{2} while the sum remains -4x

x^{2 }+ 13x - 17x -221= 0

=> x(x + 13) - 17 (x + 13) = 0

=> (x + 13) (x - 17) = 0

=> (x + 13) =0 OR (x - 17) = 0

=> x = - 13 OR x = + 17

**=> So, by factorization, we solve and find that the roots of the given equation are - 13 , + 17**

### Example 63: Using the factorization method solve the quadratic equation: x^{2} -15x +36= 0

Let's rewrite this equation as: x^{2} -15x +36= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +36x^{2} while the sum remains -15x

x^{2 }- 3x - 12x +36= 0

=> x(x - 3) - 12 (x - 3) = 0

=> (x - 3) (x - 12) = 0

=> (x - 3) =0 OR (x - 12) = 0

=> x = + 3 OR x = + 12

**=> So, by factorization, we solve and find that the roots of the given equation are + 3 , + 12**

### Example 64: Using the factorization method solve the quadratic equation: x^{2} -7x -408= 0

Let's rewrite this equation as: x^{2} -7x -408= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -408x^{2} while the sum remains -7x

x^{2 }- 24x + 17x -408= 0

=> x(x - 24) + 17 (x - 24) = 0

=> (x - 24) (x + 17) = 0

=> (x - 24) =0 OR (x + 17) = 0

=> x = + 24 OR x = - 17

**=> So, by factorization, we solve and find that the roots of the given equation are + 24 , - 17**

### Example 65: Using the factorization method solve the quadratic equation: x^{2} -23x +112= 0

Let's rewrite this equation as: x^{2} -23x +112= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +112x^{2} while the sum remains -23x

x^{2 }- 7x - 16x +112= 0

=> x(x - 7) - 16 (x - 7) = 0

=> (x - 7) (x - 16) = 0

=> (x - 7) =0 OR (x - 16) = 0

=> x = + 7 OR x = + 16

**=> So, by factorization, we solve and find that the roots of the given equation are + 7 , + 16**

### Example 66: Using the factorization method solve the quadratic equation: x^{2} +11x +24= 0

Let's rewrite this equation as: x^{2} +11x +24= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +24x^{2} while the sum remains +11x

x^{2 }+ 3x + 8x +24= 0

=> x(x + 3) + 8 (x + 3) = 0

=> (x + 3) (x + 8) = 0

=> (x + 3) =0 OR (x + 8) = 0

=> x = - 3 OR x = - 8

**=> So, by factorization, we solve and find that the roots of the given equation are - 3 , - 8**

### Example 67: Using the factorization method solve the quadratic equation: x^{2} +22x +105= 0

Let's rewrite this equation as: x^{2} +22x +105= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +105x^{2} while the sum remains +22x

x^{2 }+ 15x + 7x +105= 0

=> x(x + 15) + 7 (x + 15) = 0

=> (x + 15) (x + 7) = 0

=> (x + 15) =0 OR (x + 7) = 0

=> x = - 15 OR x = - 7

**=> So, by factorization, we solve and find that the roots of the given equation are - 15 , - 7**

### Example 68: Using the factorization method solve the quadratic equation: x^{2} -6x -432= 0

Let's rewrite this equation as: x^{2} -6x -432= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -432x^{2} while the sum remains -6x

x^{2 }+ 18x - 24x -432= 0

=> x(x + 18) - 24 (x + 18) = 0

=> (x + 18) (x - 24) = 0

=> (x + 18) =0 OR (x - 24) = 0

=> x = - 18 OR x = + 24

**=> So, by factorization, we solve and find that the roots of the given equation are - 18 , + 24**

### Example 69: Using the factorization method solve the quadratic equation: x^{2} +21x +38= 0

Let's rewrite this equation as: x^{2} +21x +38= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +38x^{2} while the sum remains +21x

x^{2 }+ 19x + 2x +38= 0

=> x(x + 19) + 2 (x + 19) = 0

=> (x + 19) (x + 2) = 0

=> (x + 19) =0 OR (x + 2) = 0

=> x = - 19 OR x = - 2

**=> So, by factorization, we solve and find that the roots of the given equation are - 19 , - 2**

### Example 70: Using the factorization method solve the quadratic equation: x^{2} -10x -39= 0

Let's rewrite this equation as: x^{2} -10x -39= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -39x^{2} while the sum remains -10x

x^{2 }+ 3x - 13x -39= 0

=> x(x + 3) - 13 (x + 3) = 0

=> (x + 3) (x - 13) = 0

=> (x + 3) =0 OR (x - 13) = 0

=> x = - 3 OR x = + 13

**=> So, by factorization, we solve and find that the roots of the given equation are - 3 , + 13**

### Example 71: Using the factorization method solve the quadratic equation: x^{2} +7x +6= 0

Let's rewrite this equation as: x^{2} +7x +6= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +6x^{2} while the sum remains +7x

x^{2 }+ 1x + 6x +6= 0

=> x(x + 1) + 6 (x + 1) = 0

=> (x + 1) (x + 6) = 0

=> (x + 1) =0 OR (x + 6) = 0

=> x = - 1 OR x = - 6

**=> So, by factorization, we solve and find that the roots of the given equation are - 1 , - 6**

### Example 72: Using the factorization method solve the quadratic equation: x^{2} -22x +40= 0

Let's rewrite this equation as: x^{2} -22x +40= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +40x^{2} while the sum remains -22x

x^{2 }- 20x - 2x +40= 0

=> x(x - 20) - 2 (x - 20) = 0

=> (x - 20) (x - 2) = 0

=> (x - 20) =0 OR (x - 2) = 0

=> x = + 20 OR x = + 2

**=> So, by factorization, we solve and find that the roots of the given equation are + 20 , + 2**

### Example 73: Using the factorization method solve the quadratic equation: x^{2} -10x -299= 0

Let's rewrite this equation as: x^{2} -10x -299= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -299x^{2} while the sum remains -10x

x^{2 }- 23x + 13x -299= 0

=> x(x - 23) + 13 (x - 23) = 0

=> (x - 23) (x + 13) = 0

=> (x - 23) =0 OR (x + 13) = 0

=> x = + 23 OR x = - 13

**=> So, by factorization, we solve and find that the roots of the given equation are + 23 , - 13**

### Example 74: Using the factorization method solve the quadratic equation: x^{2} +17x +16= 0

Let's rewrite this equation as: x^{2} +17x +16= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +16x^{2} while the sum remains +17x

x^{2 }+ 16x + 1x +16= 0

=> x(x + 16) + 1 (x + 16) = 0

=> (x + 16) (x + 1) = 0

=> (x + 16) =0 OR (x + 1) = 0

=> x = - 16 OR x = - 1

**=> So, by factorization, we solve and find that the roots of the given equation are - 16 , - 1**

### Example 75: Using the factorization method solve the quadratic equation: x^{2} -2x -99= 0

Let's rewrite this equation as: x^{2} -2x -99= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -99x^{2} while the sum remains -2x

x^{2 }+ 9x - 11x -99= 0

=> x(x + 9) - 11 (x + 9) = 0

=> (x + 9) (x - 11) = 0

=> (x + 9) =0 OR (x - 11) = 0

=> x = - 9 OR x = + 11

**=> So, by factorization, we solve and find that the roots of the given equation are - 9 , + 11**

### Example 76: Using the factorization method solve the quadratic equation: x^{2} -29x +120= 0

Let's rewrite this equation as: x^{2} -29x +120= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +120x^{2} while the sum remains -29x

x^{2 }- 5x - 24x +120= 0

=> x(x - 5) - 24 (x - 5) = 0

=> (x - 5) (x - 24) = 0

=> (x - 5) =0 OR (x - 24) = 0

=> x = + 5 OR x = + 24

**=> So, by factorization, we solve and find that the roots of the given equation are + 5 , + 24**

### Example 77: Using the factorization method solve the quadratic equation: x^{2} +14x -147= 0

Let's rewrite this equation as: x^{2} +14x -147= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -147x^{2} while the sum remains +14x

x^{2 }- 7x + 21x -147= 0

=> x(x - 7) + 21 (x - 7) = 0

=> (x - 7) (x + 21) = 0

=> (x - 7) =0 OR (x + 21) = 0

=> x = + 7 OR x = - 21

**=> So, by factorization, we solve and find that the roots of the given equation are + 7 , - 21**

### Example 78: Using the factorization method solve the quadratic equation: x^{2} +8x -209= 0

Let's rewrite this equation as: x^{2} +8x -209= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -209x^{2} while the sum remains +8x

x^{2 }- 11x + 19x -209= 0

=> x(x - 11) + 19 (x - 11) = 0

=> (x - 11) (x + 19) = 0

=> (x - 11) =0 OR (x + 19) = 0

=> x = + 11 OR x = - 19

**=> So, by factorization, we solve and find that the roots of the given equation are + 11 , - 19**

### Example 79: Using the factorization method solve the quadratic equation: x^{2} +5x -36= 0

Let's rewrite this equation as: x^{2} +5x -36= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -36x^{2} while the sum remains +5x

x^{2 }- 4x + 9x -36= 0

=> x(x - 4) + 9 (x - 4) = 0

=> (x - 4) (x + 9) = 0

=> (x - 4) =0 OR (x + 9) = 0

=> x = + 4 OR x = - 9

**=> So, by factorization, we solve and find that the roots of the given equation are + 4 , - 9**

### Example 80: Using the factorization method solve the quadratic equation: x^{2} -14x -240= 0

Let's rewrite this equation as: x^{2} -14x -240= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -240x^{2} while the sum remains -14x

x^{2 }+ 10x - 24x -240= 0

=> x(x + 10) - 24 (x + 10) = 0

=> (x + 10) (x - 24) = 0

=> (x + 10) =0 OR (x - 24) = 0

=> x = - 10 OR x = + 24

**=> So, by factorization, we solve and find that the roots of the given equation are - 10 , + 24**

### Example 81: Using the factorization method solve the quadratic equation: x^{2} -42x +432= 0

Let's rewrite this equation as: x^{2} -42x +432= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +432x^{2} while the sum remains -42x

x^{2 }- 24x - 18x +432= 0

=> x(x - 24) - 18 (x - 24) = 0

=> (x - 24) (x - 18) = 0

=> (x - 24) =0 OR (x - 18) = 0

=> x = + 24 OR x = + 18

**=> So, by factorization, we solve and find that the roots of the given equation are + 24 , + 18**

### Example 82: Using the factorization method solve the quadratic equation: x^{2} +0x -169= 0

Let's rewrite this equation as: x^{2} +0x -169= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -169x^{2} while the sum remains +0x

x^{2 }- 13x + 13x -169= 0

=> x(x - 13) + 13 (x - 13) = 0

=> (x - 13) (x + 13) = 0

=> (x - 13) =0 OR (x + 13) = 0

=> x = + 13 OR x = - 13

**=> So, by factorization, we solve and find that the roots of the given equation are + 13 , - 13**

### Example 83: Using the factorization method solve the quadratic equation: x^{2} -19x +78= 0

Let's rewrite this equation as: x^{2} -19x +78= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +78x^{2} while the sum remains -19x

x^{2 }- 13x - 6x +78= 0

=> x(x - 13) - 6 (x - 13) = 0

=> (x - 13) (x - 6) = 0

=> (x - 13) =0 OR (x - 6) = 0

=> x = + 13 OR x = + 6

**=> So, by factorization, we solve and find that the roots of the given equation are + 13 , + 6**

### Example 84: Using the factorization method solve the quadratic equation: x^{2} -16x +63= 0

Let's rewrite this equation as: x^{2} -16x +63= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +63x^{2} while the sum remains -16x

x^{2 }- 9x - 7x +63= 0

=> x(x - 9) - 7 (x - 9) = 0

=> (x - 9) (x - 7) = 0

=> (x - 9) =0 OR (x - 7) = 0

=> x = + 9 OR x = + 7

**=> So, by factorization, we solve and find that the roots of the given equation are + 9 , + 7**

### Example 85: Using the factorization method solve the quadratic equation: x^{2} -7x -170= 0

Let's rewrite this equation as: x^{2} -7x -170= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -170x^{2} while the sum remains -7x

x^{2 }+ 10x - 17x -170= 0

=> x(x + 10) - 17 (x + 10) = 0

=> (x + 10) (x - 17) = 0

=> (x + 10) =0 OR (x - 17) = 0

=> x = - 10 OR x = + 17

**=> So, by factorization, we solve and find that the roots of the given equation are - 10 , + 17**

### Example 86: Using the factorization method solve the quadratic equation: x^{2} +18x -19= 0

Let's rewrite this equation as: x^{2} +18x -19= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -19x^{2} while the sum remains +18x

x^{2 }+ 19x - 1x -19= 0

=> x(x + 19) - 1 (x + 19) = 0

=> (x + 19) (x - 1) = 0

=> (x + 19) =0 OR (x - 1) = 0

=> x = - 19 OR x = + 1

**=> So, by factorization, we solve and find that the roots of the given equation are - 19 , + 1**

### Example 87: Using the factorization method solve the quadratic equation: x^{2} +7x +6= 0

Let's rewrite this equation as: x^{2} +7x +6= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +6x^{2} while the sum remains +7x

x^{2 }+ 1x + 6x +6= 0

=> x(x + 1) + 6 (x + 1) = 0

=> (x + 1) (x + 6) = 0

=> (x + 1) =0 OR (x + 6) = 0

=> x = - 1 OR x = - 6

**=> So, by factorization, we solve and find that the roots of the given equation are - 1 , - 6**

### Example 88: Using the factorization method solve the quadratic equation: x^{2} +23x +130= 0

Let's rewrite this equation as: x^{2} +23x +130= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +130x^{2} while the sum remains +23x

x^{2 }+ 13x + 10x +130= 0

=> x(x + 13) + 10 (x + 13) = 0

=> (x + 13) (x + 10) = 0

=> (x + 13) =0 OR (x + 10) = 0

=> x = - 13 OR x = - 10

**=> So, by factorization, we solve and find that the roots of the given equation are - 13 , - 10**

### Example 89: Using the factorization method solve the quadratic equation: x^{2} -17x +66= 0

Let's rewrite this equation as: x^{2} -17x +66= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +66x^{2} while the sum remains -17x

x^{2 }- 11x - 6x +66= 0

=> x(x - 11) - 6 (x - 11) = 0

=> (x - 11) (x - 6) = 0

=> (x - 11) =0 OR (x - 6) = 0

=> x = + 11 OR x = + 6

**=> So, by factorization, we solve and find that the roots of the given equation are + 11 , + 6**

### Example 90: Using the factorization method solve the quadratic equation: x^{2} +16x +39= 0

Let's rewrite this equation as: x^{2} +16x +39= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +39x^{2} while the sum remains +16x

x^{2 }+ 3x + 13x +39= 0

=> x(x + 3) + 13 (x + 3) = 0

=> (x + 3) (x + 13) = 0

=> (x + 3) =0 OR (x + 13) = 0

=> x = - 3 OR x = - 13

**=> So, by factorization, we solve and find that the roots of the given equation are - 3 , - 13**

### Example 91: Using the factorization method solve the quadratic equation: x^{2} -15x +50= 0

Let's rewrite this equation as: x^{2} -15x +50= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +50x^{2} while the sum remains -15x

x^{2 }- 10x - 5x +50= 0

=> x(x - 10) - 5 (x - 10) = 0

=> (x - 10) (x - 5) = 0

=> (x - 10) =0 OR (x - 5) = 0

=> x = + 10 OR x = + 5

**=> So, by factorization, we solve and find that the roots of the given equation are + 10 , + 5**

### Example 92: Using the factorization method solve the quadratic equation: x^{2} +7x +12= 0

Let's rewrite this equation as: x^{2} +7x +12= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +12x^{2} while the sum remains +7x

x^{2 }+ 4x + 3x +12= 0

=> x(x + 4) + 3 (x + 4) = 0

=> (x + 4) (x + 3) = 0

=> (x + 4) =0 OR (x + 3) = 0

=> x = - 4 OR x = - 3

**=> So, by factorization, we solve and find that the roots of the given equation are - 4 , - 3**

### Example 93: Using the factorization method solve the quadratic equation: x^{2} -23x -24= 0

Let's rewrite this equation as: x^{2} -23x -24= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -24x^{2} while the sum remains -23x

x^{2 }- 24x + 1x -24= 0

=> x(x - 24) + 1 (x - 24) = 0

=> (x - 24) (x + 1) = 0

=> (x - 24) =0 OR (x + 1) = 0

=> x = + 24 OR x = - 1

**=> So, by factorization, we solve and find that the roots of the given equation are + 24 , - 1**

### Example 94: Using the factorization method solve the quadratic equation: x^{2} -24x +135= 0

Let's rewrite this equation as: x^{2} -24x +135= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +135x^{2} while the sum remains -24x

x^{2 }- 9x - 15x +135= 0

=> x(x - 9) - 15 (x - 9) = 0

=> (x - 9) (x - 15) = 0

=> (x - 9) =0 OR (x - 15) = 0

=> x = + 9 OR x = + 15

**=> So, by factorization, we solve and find that the roots of the given equation are + 9 , + 15**

### Example 95: Using the factorization method solve the quadratic equation: x^{2} -35x +300= 0

Let's rewrite this equation as: x^{2} -35x +300= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +300x^{2} while the sum remains -35x

x^{2 }- 20x - 15x +300= 0

=> x(x - 20) - 15 (x - 20) = 0

=> (x - 20) (x - 15) = 0

=> (x - 20) =0 OR (x - 15) = 0

=> x = + 20 OR x = + 15

**=> So, by factorization, we solve and find that the roots of the given equation are + 20 , + 15**

### Example 96: Using the factorization method solve the quadratic equation: x^{2} -40x +396= 0

Let's rewrite this equation as: x^{2} -40x +396= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +396x^{2} while the sum remains -40x

x^{2 }- 22x - 18x +396= 0

=> x(x - 22) - 18 (x - 22) = 0

=> (x - 22) (x - 18) = 0

=> (x - 22) =0 OR (x - 18) = 0

=> x = + 22 OR x = + 18

**=> So, by factorization, we solve and find that the roots of the given equation are + 22 , + 18**

### Example 97: Using the factorization method solve the quadratic equation: x^{2} -6x -391= 0

Let's rewrite this equation as: x^{2} -6x -391= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -391x^{2} while the sum remains -6x

x^{2 }+ 17x - 23x -391= 0

=> x(x + 17) - 23 (x + 17) = 0

=> (x + 17) (x - 23) = 0

=> (x + 17) =0 OR (x - 23) = 0

=> x = - 17 OR x = + 23

**=> So, by factorization, we solve and find that the roots of the given equation are - 17 , + 23**

### Example 98: Using the factorization method solve the quadratic equation: x^{2} +18x +17= 0

Let's rewrite this equation as: x^{2} +18x +17= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is +17x^{2} while the sum remains +18x

x^{2 }+ 1x + 17x +17= 0

=> x(x + 1) + 17 (x + 1) = 0

=> (x + 1) (x + 17) = 0

=> (x + 1) =0 OR (x + 17) = 0

=> x = - 1 OR x = - 17

**=> So, by factorization, we solve and find that the roots of the given equation are - 1 , - 17**

### Example 99: Using the factorization method solve the quadratic equation: x^{2} -5x -14= 0

Let's rewrite this equation as: x^{2} -5x -14= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -14x^{2} while the sum remains -5x

x^{2 }+ 2x - 7x -14= 0

=> x(x + 2) - 7 (x + 2) = 0

=> (x + 2) (x - 7) = 0

=> (x + 2) =0 OR (x - 7) = 0

=> x = - 2 OR x = + 7

**=> So, by factorization, we solve and find that the roots of the given equation are - 2 , + 7**

### Example 100: Using the factorization method solve the quadratic equation: x^{2} -8x -345= 0

Let's rewrite this equation as: x^{2} -8x -345= 0

A key step here is to factorize the quadratic polynomial expression properly. The middle term needs to be split into two terms whose product is -345x^{2} while the sum remains -8x

x^{2 }+ 15x - 23x -345= 0

=> x(x + 15) - 23 (x + 15) = 0

=> (x + 15) (x - 23) = 0

=> (x + 15) =0 OR (x - 23) = 0

=> x = - 15 OR x = + 23

**=> So, by factorization, we solve and find that the roots of the given equation are - 15 , + 23**