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: