Introduction To Quadratic Equations And Methods, Definition Important Topics For JEE 2024

Introduction To Quadratic Equations

Introduction To Quadratic Equations : We studies about different type of polynomials like linear polynomial ( ax + b ), a ≠ 0 quadratic polynomial  ( ax ² + bx + c ) , a ≠ 0, cubic polynomial ( ax ³ + bx ² + cx + d ) , a ≠ 0 etc. When we equate a quadratic polynomial to zero, we get a quadratic equation i.e. ax ² + bx + c = 0 a ≠ 0 and a , b , c are real numbers and x is a variable is called quadratic equation.

For example

Problem For Practice (Definition Of Quadratic Equations)

1. Find which of the following is/are quadratic equation?

(i) x + 1 = 0

(ii)

(iii)

(iv)

(v)

Uses Of Quadratic Equations In Real Life

Uses Of Quadratic Equations In Real Life : We used quadratic equation in many real life situations for example your school wants to make a Indore badminton court having area 300 square meter with its length one meter more than twice is breadth. What should be length and breath of the hall.

Suppose breath of the hall is x meters so its length should be 2 x + 1.

So area of court is

This represent a quadratic equation. Breath of court should be satisfy the equation.

History Behind Quadratic Equation

History Behind Quadratic Equation : Our knowledge Babylonians were first to solve quadratic equations. They knew how to find two positive numbers with a given positive sum and given a positive product this type problem related to quadratic equations. Solving of quadratic equation, in general form is after created goes to ancient Indian mathematicians. Brahmagupta gave a explicit formula to solve quadratic equation. Later Sridharacharya derived a formula also known as Sridharacharya method which we are going to use in this chapter many times.

Roots Of Quadric Equation

Roots Of Quadric Equation : Let us consider a quadratic equation

x ² +3x+2=0

If we put in L.H.S then we get

Then we say that is the roots of the equation x ² +3x+2=0

In other words we say that is a solution of the quadratic equation.

Note: Zeros of quadratic polynomial and the roots of quadratic equation are same.

Factorization Method To Solve Quadratic Equation

Factorization Method To Solve Quadratic Equation : Let consider a quadratic equation

Step 1: Find two numbers such that their product is equal to ac and their sum equal to b

Step 2: Split the middle term using these two numbers.

Step 3: Take the common factor and simplify

Factorization Method To Solve Quadratic Equation Example 1

Factorization Method To Solve Quadratic Equation Example 1 : Solve the quadratic equation using factorization method

Sol. Give quadratic equation

Step 1: Find two number such that their product is 14 and their sum is 9 so these two numbers are 2 and 7

Step 2: Now split the middle term in two numbers

Step 3: Now take common factor

or

or

Hence x = –2 and x = –7 are roots of given quadratic equation.

Example 2:

Solve the equation by factorisation method

Sol. Consider the given equation

Now

⇒

⇒

⇒ or

Hence and x = 4 are solutions of given equation.

Problem For Practice (Factorization Method)

1. Find the solutions of the following quadratic equation using factorization method

(i)

(ii)

(iii)

(iv)

(v)

Quadratic Formula

Quadratic Formula : For quadratic equation , a , b , c ∈ R

or

where known as discriminant denote by D

Proof of quadratic formula using completing square method

Let consider the quadratic equation

Divide the equation by coefficient of x ² i.e. a .

Subtract c a from both side of this equation

Now use complete square method adding both side

⇒

⇒

⇒

⇒

⇒

or

Example 1:

Find the roots of quadratic equation

Sol.

Consider the given quadratic equation

Compare with standard equation

Now a = 2, b = 4, c = 1

Now using quadratic formula

or

or

Hence roots of quadratic equation are

and

Example 2:

Find the roots of quadratic equation

Sol.

Consider the given equation.

Compare with standard equation

We find that

a = 1, b = 4, c = –5

Now using quadratic formula

Putting values of a , b and c

So, or

x = 1 or x = –5

x = 1 and x = –5 are roots of equation.

Example 3:

Find the roots of quadratic equation

Sol.

Consider the given equation

Compare with standard equation

find a , b , c

a = 1, b = 1, c = 1

Using quadratic formula

or

Hence roots of equation are

and

Problem For Practice (Quadratic Formula)

1. Find the roots of following quadratic equations

(i)

(ii)

(iii)

(iv)