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: ax2+bx+c=0 (a is not equal to 0); and solving for x gives
x = (-b + √D ) / 2a
and
x = (-b - √D ) / 2a
Where D is the Discriminant and D = b2-4ac
The Discriminant
The term b2-4ac is called the Discriminant, and is denoted usually by the symbol ∆ or the letter ‘D’.
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 :
ax4+bx2+c=0; substitute x2=y, to form a quadratic equation in y.
3x+√x-2=0; substitute √x=y to form a quadratic equation in y.
x+√(x-4)=4; transport x to RHS and then square both sides to get a quadratic equation.
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) in the original equation, then x has to be greater than 1, as square root cannot be negative.
Also, if you have a step like: (x-z)(f(x))=(x-z)(g(x)) then, instead of just dividing both sides by (x-z), you write x=z as one of the solutions.
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. (Right now, we are talking about the degree being an integer)
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: ax5+bx4+cx3+dx2+ex+f=0. Then
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 xn
Here are some of the examples and problems solved in the tutorial :
Q: Solve x2-5x+6=0
Q: Solve 27x2-10x+1=0
Q: Solve x2+4ix-4=0
Q: Solve x2-(7-i)x+(18-i)=0.
Q: Write an equation whose roots are 13,89.
Q: If α,β are roots of ax2+bx+c=0, find the value of 1/α+1/β; α2+β2
Q: If α,β are roots of x2-a(x+1)-c=0, find the value of (1+α)(1+β)
Q: If α,β are roots of ax2+bx+c=0, write the equation whose roots are 1/α4 and -1/β4.
Q: If 3+√5 is a root of x2+bx+c=0, find the values of b and c, given that b and c are real.
Q: The equation x2-kx+k+2=0 will have equal roots for what value(s) of k?
Q: Find the number of real roots of the equation (x2+2x)2-(x+1)2-55=0.
Q: Find the number of solutions of x2+|x-1|=1
Q: Find the value of λ such that x2+2x+3λ=0 and 2x2+3x+5λ=0 have a non zero common root.
Q: If α,β are roots of x2+px+1=0 and γ,δ are the roots of x2+qx+1=0, evaluate:
(α-γ)(α+δ)(β-γ)(β+δ).
Q: The real numbers x1, x2, x3 satisfying the equation x3 -x2 + bx + c =0 are in AP. Find the intervals in which b and c lie.
Q: Find the equation whose roots are cube of the roots of the equation ax3+bx2+cx+d=0.
