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Solving Quadratic Equations

22.3 Solution by Completing the Square

We have seen that expressions of the form :

a2x2 − b2

are known as differences of squares and can be factorised as follows:

(ax − b)(ax + b).

This simple factorisation leads to another technique to solve quadratic equations known as
completing the square.

We demonstrate with a simple example , by trying to solve for x in :

x2 − 2x − 1 = 0.(22.1)

We cannot easily find factors of this term, but the first two terms look similar to the first two
terms of the perfect square :

(x − 1)2 = x2 − 2x + 1.

However, we can cheat and create a perfect square by adding 2 to both sides of the equation in
(22.1) as:

Now we know that:

which means that:

(x − 1)2 − 2

is a difference of squares. Therefore we can write:

he solution to x2 − 2x − 1 = 0 is then:

or

This means or . This example demonstrates the use of completing the
square to solve a quadratic equation .

Method : Solving Quadratic Equations by Completing the Square

1. Write the equation in the form ax2 + bx + c = 0. e.g. x2 + 2x . 3 = 0

2. Take the constant over to the right hand side of the equation. e.g. x2 + 2x = 3

3. If necessary, make the coefficient of the x 2 term = 1, by dividing through by the existing
coefficient.

4. Take half the coefficient of the x term, square it and add it to both sides of the equation.
e.g. in x2 + 2x = 3, half of the x term is 1. 11 = 1. Therefore we add 1 to both sides to
get: x2 + 2x + 1 = 3 + 1.

5. Write the left hand side as a perfect square: (x + 1)2 - 4 = 0

6. You should then be able to factorise the equation in terms of difference of squares and
then solve for x: (x + 1 - 2)(x + 1 + 2) = 0

Worked Example 106: Solving Quadratic Equations by Completing the
Square


Question: Solve:

x2 − 10x − 11 = 0

by completing the square

Answer

Step 1 : Write the equation in the form ax2 + bx + c = 0

x2 − 10x − 11 = 0

Step 2 : Take the constant over to the right hand side of the equation


x2 − 10x = 11

Step 3 : Check that the coefficient of the x2 term is 1.

The coefficient of the x2 term is 1.

Step 4 : Take half the coefficient of the x term, square it and add it to both
sides

The coefficient of the x term is . Therefore:

x2 − 10x + 25 = 11 + 25

Step 5 : Write the left hand side as a perfect square

(x − 5)2 − 36 = 0

Step 6 : Factorise equation as difference of squares

(x − 5)2 − 36 = 0

[(x − 5) + 6][(x − 5) − 6] = 0

Step 7 : Solve for the unknown value

[x + 1][x − 11] = 0

x = −1 or x = 11
 
Worked Example 107: Solving Quadratic Equations by Completing the
Square


Question: Solve:

2x2 − 8x − 16 = 0

by completing the square

Answer

Step 1 : Write the equation in the form ax2 + bx + c = 0

2x2 − 8x − 16 = 0

Step 2 : Take the constant over to the right hand side of the equation

2x2 − 8x = 16

Step 3 : Check that the coefficient of the x2 term is 1.

The coefficient of the x2 term is 2. Therefore, divide both sides by 2:

x2 − 4x = 8

Step 4 : Take half the coefficient of the x term, square it and add it to both
sides


The coefficient of the x term is . Therefore:

x2 − 4x + 4 = 8 + 4

Step 5 : Write the left hand side as a perfect square

(x − 2)2 − 12 = 0

Step 6 : Factorise equation as difference of squares



Step 7 : Solve for the unknown value



Step 8 : The last three steps can also be done in a different the way

Leave left hand side written as a perfect square

(x − 2)2= 12

Step 9 : Take the square root on both sides of the equation



Step 10 : Solve for x

Therefore or

Compare to answer in step 7.
 

 

Exercise: Solution by Completing the Square

Solve the following equations by completing the square:

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