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Algebra

Quadratic Equations

ALL MODULES

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

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, 2

etc.

2x^{3} + 4x^{2} + 5, 5y^{53} + 2, x - 5

and

84

are polynomials. Note that

84

is a polynomial as the power of the variable is

0

(which is a non-negative integer), i.e.

84 = 84x^{0}

\dfrac{2}{x} = 2x^{-1}, 2 \sqrt{x} + 5

and

\sqrt[5]{x} - \sqrt[3]{x - 5} + 7

are not polynomials as the powers are not non-negative integers.

Polynomials can have any number of variables. For instance,

x^{3}y - xy^{2} + 2

is a polynomial with two variables.

A polynomial of a single variable can be represented as f

(x) = a_{0} + a_{1}x^{1} + a_{2}x^{2} + ... + a_{n}x^{n}

, where

x

is a variable,

a_{0}

is a constant and

a_{1}, a_{2}, ... , a_{n}

are coefficients of

x^{1}, x^{2}, ... , x^{n}

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

x

2x^{3}yz = 3

Degree of

2x^{3}yz = 3 + 1 + 1 = 5

Degree of

5x^{4}y^{2} = 4 + 2 = 6

Degree of

2x^{3}yz - 5x^{4}y^{2} + 2 = 6

Equations with degrees

1, 2, 3

and

4

are called Linear Equation, Quadratic Equation, Cubic Equation and Biquadratic equation respectively. This lesson covers equations with degrees greater than

1

3. Quadratic Expression & Equations

A quadratic expression with

1

variable is of the form

ax^{2} + bx + c

, where

a \ne 0

.

This can also be written as a function.

f(x) = ax^{2} + bx + c

, where

a \ne 0

.

Roots of an expression are the

x

-values (values of

x

) where the expression equates to

0

.

Where

\alpha

and

\beta

are the roots, the quadratic equation formed is

(x - \alpha)

(x - \beta) = 0

Example 1

Form the quadratic equations for the following sets of roots
(I)

(2, 3)

(II)

(3, - 4)

(III)

\left(\dfrac{2}{3}, -\dfrac{1}{2} \right)

Solution

Case I:

(x - 2)(x - 3) = 0

⇒

x^{2} - 5x + 6 = 0

Case II:

(x - 3) (x - (-4)) = 0

⇒

(x -3)(x + 4) = 0

⇒

x^{2} + x - 12 = 0

Case III:

\left(x - \dfrac{2}{3} \right) \left(x - \left(-\dfrac{1}{2}\right) \right) = 0

⇒

\left(\dfrac{3x - 2}{3} \right) \left(\dfrac{2x + 1}{2} \right) = 0

⇒

(3x - 2) (2x + 1) = 0

⇒

6x^{2} - x - 2 = 0

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