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**Quadratic Equations, Cubic and Higher Order Equations**

**Introduction **

Quadratic equations are those equations which can be written in the form f(x)=0 where f(x) is a second degree polynomial. General form of a Quadratic equation is: ax^{2}

x = (-b + **√D** ) / 2a

and

x = (-b - **√D **) / 2a

Where D is the Discriminant and* D = b*^{2}

**The Discriminant**

The term b^{2}

**Roots of a Quadratic Equation : Are they real/unreal, equal or unequal ?**

If D>0, the equation has real and unequal roots, if D=0, the equation has real and equal roots (also called real repeated roots), and if D<0, the equation has unreal roots, occurring as conjugate pairs. That is if one root is of

the form u+iv, the other root would be u-iv. If α and β are roots of a Quadratic equation, then

o The equation can be written as : a(x-α)(x-β)=0

o The equation could also be written as x-(α+β)+αβ=0

**Using Substitution to convert equations to Quadratic Form**

A Quadratic equation could be solved by factorization, or by using the direct formula written above. Certain equations are not quadratic, but can be reduced to a quadratic form by certaing substitutions. In such cases, applying the right form of substitution yields the required solutions.

Example :

**ax**^{4}^{2}^{2}

**3x+√x-2=0; **

**x+√(x-4)=4; **

Remember, when using substitutions, be sure that you solve for the original variable, and that the solution does not violate any constraints. For instance, if you have **√(x-1) **

Also, if you have a step like: *(x-z)(f(x))=(x-z)(g(x))*

**Dealing with Cubic and Higher Order Equations **

A cubic equation is of the form f(x)=0, where f(x) is a degree 3 polynomial. The general form of a cubic equation is ax3+bx2+cx+d=0, where a is not equal to 0. If α, β, γ are roots of the equation, then equation could be written as:

**a(x-α)(x-β)(x-γ)=0, or also as**

**x3-(α+β+γ)x2+(αβ+βγ+γα)x-(αβγ)=0**

Thus, we have

α+β+γ=-b/a

αβ+βγ+γα=c/a

αβγ=-d/a

A quadratic equation may have all repeated real roots, two repeated and one distinct real root,

one distinct real and two conjugate unreal roots, all distinct real roots.

**Higher Order equations - Relationship between roots**

Equations of the type f(x)=0 where degree of f(x) is greater than 3 are generally termed as **Higher Order equations**

The number of roots of an equation with real coefficients is equal to the degree of f(x). The way to solve a higher order equation is by factorization, or by using the factor theorem, or by reducing it to one of the lower order equations.

The factor theorem is: (x-a) is a factor of f(x) if f(a)=0.

The relationship between the coefficients and the roots can be explained by an example. Consider a fifth degree equation:** ax**^{5}^{4}^{3}^{2}

Sum of roots = -b/a

Sum of pairwise products of roots=c/a

Sum of products of roots taken three at a time = -d/a

Sum of products of roots taken four at a time = e/a

Product of roots=-f/a

In general, for a n degree polynomial equation in variable x, sum of products of roots taken m at a time is (-1)^{m}.Coefficient of xn-m/Coefficient of x^{n}

**Here are some of the examples and problems solved in the tutorial :**

- Solving basic quadratic equations
- Using factorization or formula method for finding the roots
- Will roots be equal unequal or complex? Identifying this using the discriminant (D < 0, D = 0, D > 0 and so on)
- being able to plot the parabolic curve representing the quadratic equation and identifying roots from the plots or graph
- Write an equation whose roots are provided
- If α,β are roots of a quadratic equation find their sum and product.
- Find the number of real roots of an equation.
- Find the number of solutions of a given equation.
- Questions related to simple cubic equations.