Factorisation and Solving Quadratic Equations by Completing the Square

(a + b)^{2} = a^{2} + 2ab + b^{2}

Notice the 2 which is multiplying a and b above.

We will now apply similar logic to the examples below:

__Example 1:__

Use completing the square to factorise and solve for X in: X^{2} + 4X = 0

4X = 2 x 2 x X

So we are dividing 4 by 2 to get 2.

We will now square 2.

2^{2} = 4

To enable us factorise, we will now add 4 and subtract 4 to maintain the value of the equation:

X^{2} + 4X + 4 - 4 = 0

Therefore:

(X + 2)^{2} - 4 = 0

Factorising, we will now have:

(X + 2)^{2} - 4 = 0

Add 4 to both sides:

(X + 2)^{2} = 4

Apply square root to both sides:

X + 2 = ±2

Therefore:

X = +2 - 2 = 0

OR

X = -2 -2 = - 4

__Example 2:__

Use completing the square to factorise and solve for X in: X^{2} + 6X = 0

6 ÷ 2 = 3

3^{2} = 9

Therefore:

X^{2} + 6X + 9 - 9 = 0

(X + 3)^{2} - 9 = 0

(X + 3)^{2} = 9

X + 3 = ±3

Therefore:

X + 3 = + 3

X = + 3 - 3 = 0

OR

X = - 3 - 3 = -6

__Example 3:__

Write x^{2} + 10x + 3 in the form (x + a)^{2} + b, where a and b are constants

10 ÷ 2 = 5

5^{2} = 25

x^{2} + 10x + 25 - 25 + 3

(x + 5)^{2} - 25 + 3

(x + 5)^{2} - 22

Now try the these: