Syllabus

carry out the process of completing the square for a quadratic polynomial ax^2 + bx + c and use a completed square form

• e.g. to locate the vertex of the graph of y = ax^2 + bx + c or to sketch the graph

Steps

Completing the square involves transforming a quadratic expression of the form ax^2 + bx + c into a new expression in the form (a(x-h)^2 + k), where hand k are constants. The steps to complete the square are as follows:

Example: Consider the expression 2x^2 + 4x + 3.

1. Bring out the coefficient of x^2 (a).

2(x^2 + 2x) + 3

2. Divide the coefficient of x by 2 and square this value. Add and deduct this squared value in the parenthesis.

2(x^2 + 2x + 1 - 1) + 3

3. Factor the first three items in the parenthesis as a perfect square of the form (x-h)^2.

   2((x + 1)^2 - 1) + 3

4. Move the constant term out of the parenthesis, then simplify the constant terms.

   \begin{aligned}
   &2(x + 1)^2 - 2 + 3\\
   =&2(x + 1)^2 + 1
   \end{aligned}