CSEC Mathematics: Completing the Square of a Quadratic Expression
You are likely familiar with the general form of a quadratic expression, ax² + bx + c, where
a and b are coefficients, c is a constant and x is an unknown variable. Completing the square is essentially expressing a quadratic expression in the form a (x + h)² + k.
You can calculate the values of h and k like this:

So, for a quadratic expression 6x² + 5x + 9:
h= 5/2(6)= 5/12
k= 9- (5²)/4(6)= 9 - 25/24 = 191/24
6x² + 5x + 9 expressed in the form a (x + h)² + k is:
6(x + 5/12)² + 191/24
However, you don't need to memorize those two formulas to complete the square. You can also follow a very simple procedure to do the same thing.
If we take the same expression 6x² + 5x + 9, we follow these steps to complete the square:
Step 1
Divide all terms by the coefficient of x² (a). In this equation, that is 6.
(Note: This only applies where a ≠ 1. If a = 1, then just move on to step 2)
6x²/6 + 5x/6 + 9/6
Simplified:
6 (x² + (5/6)x + 3/2)
Step 2
The b term is now 5/6. Right before the c term, we will add (b/2)², or (5/12)² and we will subtract (b/2)², or (5/12)² after the c term:
6 (x² + (5/6)x + (5/12)² + 3/2 - (5/12)²)
Simplified:
6 (x² + (5/6)x + (5/12)² + 3/2 - 25/144)
6 (x² + (5/6)x + (5/12)² + 191/144)
Step 3
Simplify/Write the first three terms (x², bx and (b/2)²) as a squared expression:
6 ((x + 5/12)² + 191/144)
Step 4
Multiply the term (191/144) left outside the squared expression by 6 so that 6 is only multiplying the squared expression. (This only applies to equations where a was not equal to 1)
6(x + 5/12)² + 191/24
And just like that, we have our answer!
If you didn't follow those steps all too well, look at the following diagram to get a better understanding:
