## Solution of a Quadratic Equation by Completing the Square

Following steps are involved in solving a quadratic equation by quadratic formula

· Consider the equation , where a ¹ 0

· Dividing throughout by ‘a’, we get

· Add and subtract , we get

,

· If taking square root of both sides, we obtain

Therefore

The Quadratic Formula: Quadratic equation, where a, b, c are real number and , has the roots as

## Problems based on Solution of a Quadratic Equation by Completing the Square with Solutions

### Problem:

Solve by completing the square :

### Solution:

(i)

Þ

Adding i.e. on both sides, we have

Þ

or

Þ

Þ

So, the solutions are

### Problem:

By using the method of completing the square, show that the equation has no real roots.

### Solution:

Þ [Dividing the equation by 2]

Þ

Adding i.e. on both sides, we have

Þ

Þ

Þ

RHS is negative but cannot be negative for any real values of x.

[ square of any real number is non-negative]

Hence, the given equation has no real roots.

### Problem:

Find the roots of the following equation

by the method of completing the square.

### Solution:

We have,

Þ [Dividing the equation by 4]

Þ

Adding on both sides we have

Þ

Þ

Þ [taking square root of both sides]

Þ

Þ

Hence, the roots are and

### Problem:

Solve the equation by the method of completing the square.

### Solution:

We have,

Þ

Adding on both sides we have

Þ

Þ

Þ

Þ

Þ

or

Þ

Hence, the roots are and 1.

### Problem:

Using quadratic formula, solve the following quadratic equation for x:

### Solution:

Comparing the given equation

with the standard quadratic equation

We have a = 1

b = –2a

And

Now Discriminant

Þ Þ

Using the quadratic formula, we get

Hence or

### Problem:

Using quadratic formula solve the following quadratic equations:

(i) (ii)

### Solution:

(i) We have,

Comparing the equation with we have

and

\

[ square of any real number is non-negative]

So, the given equation has real roots given by

and,

Hence the roots are and -1

(ii) We have,

Comparing this equation with we have

and

\

Þ

Þ

Þ

Þ

Þ [ square of any real number is non-negative]

So, the roots of the given equation are real and are given by

and,

\ Solution is and

### Problem:

Using the quadratic formula, solve the equation

### Solution:

Comparing the given equation with

We have

Discriminant

[ square of any real number is non-negative]

Hence, the given equation has real roots given by

The roots are

and

or and

or and

Hence, are the required solutions.