12.6 Completing the square — Learn

Prerequisites: expanding brackets, collecting terms, and solving equations by square roots.

Objectives: rewrite a quadratic by completing the square, solve it, and use the form to find the turning point. This covers A11 and A18 at Higher level.

Build a square

For $x^2+bx+c$, take half of $b$, square it, and add and subtract that value.

Example 1. Complete the square for $x^2+6x+2$.

Half of $6$ is $3$, and $3^2=9$.

The turning point of $y=(x+3)^2-7$ is $(-3,-7)$ because the square is smallest at $x=-3$.

Example 2. Solve $x^2+6x+2=0$.

Watch out. The half coefficient is squared: half of $6$ is $3$, and the added value is $9$. Check by expanding the completed form.

You should now be able to

• rewrite a monic quadratic in completed-square form;
• solve an equation by isolating the square and taking both square roots;
• read a turning point from completed-square form.