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Quadratic Equations

Quadratic Equations

MODULES

Basics of Polynomial & Quadratic Equations
Discriminant & Graphical Representation
Sum & Product of Roots
Factorisation Method
Formulation & Completion of Squares
Changes to Roots
Mistakes in Roots, Common Roots & Squaring
Infinite Series & Transposed
Other Types
Higher Order Equations
Synthetic Division & Remainder Theorem
Maxima, Minima & Descrates Rule
Past Questions

CONCEPTS & CHEATSHEET

Concept Revision Video

SPEED CONCEPTS

Quadratic Equations 1
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Quadratic Equations 2
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Quadratic Equations 3
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PRACTICE

Quadratic Equations : Level 1
Quadratic Equations : level 2
Quadratic Equations : level 3
ALL MODULES

CAT 2025 Lesson : Quadratic Equations - Basics of Polynomial & Quadratic Equations

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1. Introduction

An equation simply represents an equality. One or more variables with one or more powers for the variable(s) when equated forms an equation. This is usually formed with arithmetic operations (+,−,(+, -,(+,−, ×,/)\times, /)×,/) and can have constants or variable terms on both sides of an equation.

In the previous lesson we looked at linear equations in detail. In this lesson we will look at all higher order equations.

2. Polynomial

A polynomial comprises of one or more variables and constants. The powers of each of the variables have to be non-negative integers, i.e. 0,1,20, 1, 20,1,2 etc.

2x3+4x2+5,5y53+2,x−52x^{3} + 4x^{2} + 5, 5y^{53} + 2, x - 52x3+4x2+5,5y53+2,x−5 and 848484 are polynomials. Note that 848484 is a polynomial as the power of the variable is 000 (which is a non-negative integer), i.e. 84=84x084 = 84x^{0}84=84x0.

2x=2x−1,2x+5\dfrac{2}{x} = 2x^{-1}, 2 \sqrt{x} + 5x2​=2x−1,2x​+5 and x5−x−53+7\sqrt[5]{x} - \sqrt[3]{x - 5} + 75x​−3x−5​+7 are not polynomials as the powers are not non-negative integers.

Polynomials can have any number of variables. For instance, x3y−xy2+2x^{3}y - xy^{2} + 2x3y−xy2+2 is a polynomial with two variables.

A polynomial of a single variable can be represented as f(x)=a0+a1x1+a2x2+...+anxn(x) = a_{0} + a_{1}x^{1} + a_{2}x^{2} + ... + a_{n}x^{n}(x)=a0​+a1​x1+a2​x2+...+an​xn, where xxx is a variable, a0a_{0}a0​ is a constant and a1,a2,...,ana_{1}, a_{2}, ... , a_{n}a1​,a2​,...,an​ are coefficients of x1,x2,...,xnx^{1}, x^{2}, ... , x^{n}x1,x2,...,xn respectively.

2.1 Degree of a Polynomial

Degree of a variable is the power of the variable.

Degree of a term in a polynomial is the sum of degrees of variables in that term.

Degree of a polynomial is the highest degree of a term in the polynomial.

Degree of xxx in 2x3yz=32x^{3}yz = 32x3yz=3
Degree of 2x3yz=3+1+1=52x^{3}yz = 3 + 1 + 1 = 52x3yz=3+1+1=5
Degree of 5x4y2=4+2=65x^{4}y^{2} = 4 + 2 = 65x4y2=4+2=6
Degree of 2x3yz−5x4y2+2=62x^{3}yz - 5x^{4}y^{2} + 2 = 62x3yz−5x4y2+2=6

Equations with degrees 1,2,31, 2, 31,2,3 and 444 are called Linear Equation, Quadratic Equation, Cubic Equation and Biquadratic equation respectively. This lesson covers equations with degrees greater than 111.

3. Quadratic Expression & Equations

A quadratic expression with 111 variable is of the form ax2+bx+cax^{2} + bx + cax2+bx+c, where a≠0a \ne 0a=0.

This can also be written as a function. f(x)=ax2+bx+cf(x) = ax^{2} + bx + cf(x)=ax2+bx+c, where a≠0a \ne 0a=0.

Roots of an expression are the xxx-values (values of xxx) where the expression equates to 000.

Where α\alphaα and β\betaβ are the roots, the quadratic equation formed is (x−α)(x - \alpha)(x−α) (x−β)=0(x - \beta) = 0(x−β)=0

Example 1

Form the quadratic equations for the following sets of roots
(I) (2,3)(2, 3)(2,3)
(II) (3,−4)(3, - 4)(3,−4)
(III) (23,−12)\left(\dfrac{2}{3}, -\dfrac{1}{2} \right)(32​,−21​)

Solution

Case I: (x−2)(x−3)=0(x - 2)(x - 3) = 0(x−2)(x−3)=0 ⇒ x2−5x+6=0x^{2} - 5x + 6 = 0x2−5x+6=0

Case II:(x−3)(x−(−4))=0(x - 3) (x - (-4)) = 0(x−3)(x−(−4))=0 ⇒ (x−3)(x+4)=0(x -3)(x + 4) = 0(x−3)(x+4)=0

⇒ x2+x−12=0 x^{2} + x - 12 = 0x2+x−12=0

Case III: (x−23)(x−(−12))=0\left(x - \dfrac{2}{3} \right) \left(x - \left(-\dfrac{1}{2}\right) \right) = 0(x−32​)(x−(−21​))=0 ⇒ (3x−23)(2x+12)=0 \left(\dfrac{3x - 2}{3} \right) \left(\dfrac{2x + 1}{2} \right) = 0(33x−2​)(22x+1​)=0

⇒ (3x−2)(2x+1)=0 (3x - 2) (2x + 1) = 0(3x−2)(2x+1)=0 ⇒ 6x2−x−2=0 6x^{2} - x - 2 = 06x2−x−2=0


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