Completing The Square
Factorisation and Solving Quadratic Equations by Completing the Square

(a + b)2 = a2 + 2ab + b2
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: X2 + 4X = 0
4X = 2 x 2 x X
So we are dividing 4 by 2 to get 2.
We will now square 2.
22 = 4
To enable us factorise, we will now add 4 and subtract 4 to maintain the value of the equation:
X2 + 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: X2 + 6X = 0
6 ÷ 2 = 3
32 = 9
Therefore:
X2 + 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 x2 + 10x + 3 in the form (x + a)2 + b, where a and b are constants
10 ÷ 2 = 5
52 = 25
x2 + 10x + 25 - 25 + 3
(x + 5)2 - 25 + 3
(x + 5)2 - 22

Now try the these: