Quadratic Equations, Cubic and Higher Order Equations : Plots, Factorization, Formulas

Target Audience: High School Students, College Freshmen and Sophomores, Class 11/12 Students in India preparing for ICSE/ISC/CBSE and Entrance Examinations like the IIT-JEE
Main or Advanced/AIEEE
, and anyone else who needs this Tutorial as a reference!

After studying the topic, you might benefit from the MCQ Quiz over here.

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- 4ac

The Discriminant

The term b²- 4ac

How do we plot the curve on a graph paper?

For various values of 'x', evaluate the expression f(x) similar to the way in which algebraic expressions in one variable are evaluated.

There are plenty of examples for evaluating algebraic polynomials for specific values of 'x': here and here.

In the case of quadratic polynomials, we get a parabolic curve.

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

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

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

Roots can be verified using the factor theorem (Check out this tutorial specially example 6).

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

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: ax⁵ + bx⁴+ cx³ + dx² + e = 0

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 :

Factorizing Quadratic Expressions By Splitting the Middle Terms

Simple Examples: Solve Quadratic Equations via Factorization

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.

A Visual Analysis of Graphs of Quadratic Functions

Complete Tutorial with Solved Problems (To take a shot at the Companion Quiz, scroll down the current page):

MCQ Quiz/Worksheet for Quadratic Equations- test how much you know about the topic !

Your score will be e-mailed to you at the address you provide.

Quadratic Equations : Companion Quiz

Appendix: More Graph of Quadratic Functions

Introduction to Complex Numbers Introduction to Complex Numbers and iota. Argand plane and iota. Complex numbers as free vectors. N-th roots of a complex number. Notes, formulas and solved problems related to these sub-topics.	The Principle of Mathematical InductionIntroductory problems related to Mathematical Induction.	Quadratic Equations Introducing various techniques by which quadratic equations can be solved - factorization, direct formula. Relationship between roots of a quadratic equation. Cubic and higher order equations - relationship between roots and coefficients for these. Graphs and plots of quadratic equations.	Quadratic Inequalities Quadratic inequalities. Using factorization and visualization based methods.
Series and Progressions Arithmetic, Geometric, Harmonic and mixed progressions. Notes, formulas and solved problems. Sum of the first N terms. Arithmetic, Geometric and Harmonic means and the relationship between them.

Quizzes on Progressions
MCQ #1: Arithmetic Progression
MCQ #2: Geometric Progression
MCQ #3 : More on Geometric Progressions.
MCQ #4 : Harmonic Progressions.
MCQ #5: More on Harmonic Progression
MCQ #6: Mixed Progressions

Complex Numbers
MCQ #1
MCQ #2: More on Complex Numbers

Quadratic Equations
MCQ Quadratic Equations

Quadratic In-equations
MCQ Quadratic In-equations

Coordinate Geometry - Straight Lines
MCQ #1: Cartesian Planes, Straight Line Basics
MCQ #2 on Straight Lines
MCQ #3 on Straight Lines
MCQ #4 on Straight Lines

Circles
1 MCQ #1 on Circles.
2 MCQ #2 on Circles.
3 MCQ #3 on Circles.

Conic Sections- Parabola, Hyperbola, Ellipse
1 MCQ- The Basics of Conic Sections
2 MCQ on Parabola..
3 MCQ on Hyperbola
4 MCQ on Ellipses.

Probability
MCQ #1 on Basic Probability
MCQ #2: More Challenging Problems on Probability
MCQ #3- Conditional Probability and Bayes Theorem

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