# Completing the square

**Perfect squares**

Quadratic expressions can be written as perfect squares, for example:

*x*² + 2*ax* + *a*² = (*x* + *a*)²

*x*² – 2*ax* + *a*² = (*x* – *a*)²

The process of making an expression into a perfect square is called **completing the square.**

**Completing the square**

When completing the square for the following expression, adding a 9 would covert it to a perfect square.

*x*² + 6*x *(add 9)

*x*² – 6*x* = *x²* – 6*x + 9 – 9 *(if 9 is added, it must be subtracted also for both sides to be equal)

*x*² – 6*x* = (*x* – 3*)² – 9 *(this step completes the square)

Therefore the following rules apply:

**Solving quadratics by completing the square**

In circumstances where a quadratic equation cannot be solved by factorisation, then it can be solved by completing the square. For example:

*x*² – 8*x* + 5 = 0 (complete the square on the left hand side)

(*x + 4*)² – 11 = 0* *(simplify both sides)

(*x + 4*)² = 11 (square root both sides

(*x + 4*) = ±√11

*x = *±√11 -4

*x = *-0.683

*x = *-7.317 (to 3 d.p.)