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
x = (-b + √D ) / 2a
and
x = (-b - √D ) / 2a
Where D is the Discriminant and D = b2
The Discriminant
The term b2
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 :
ax422
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: ax5432
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 :
- 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.
