Completing the square

Video lesson3 worked examples

Theory

Completing the Square transforms a general quadratic $y = ax^2 + bx + c$ into the vertex form $y = (x + p)^2 + q$ .

It is a powerful technique because it works for all quadratics, even those that do not have real roots.

Once in this form, you can immediately identify the exact coordinates of the parabola's turning point, which are given by $(-p, q)$ .

The Golden Rule: $p$ is always half the coefficient of $x$ . Write the bracket $(x + p)^2$ , then subtract $p^2$ to cancel the extra it introduces, and finally add the original constant.

⚠️ Common Examiner Traps

Forgetting to subtract $p^2$ : $(x + p)^2$ expands to include an extra $p^2$ , which must be subtracted back off.
Sign of $p$ : a negative $x$ coefficient gives a negative $p$ , e.g. $x^2 - 6x$ leads to $(x - 3)^2$ .
Odd coefficients: if the $x$ coefficient is odd, $p$ is a fraction — keep it exact, do not round.
Turning point sign: from $(x + p)^2 + q$ the turning point is $(-p, q)$ — the $x$ -coordinate has the opposite sign to $p$ .

Worked examples

Example 1

Completing the Square

Express $x^2 + 8x + 5$ in the form $(x + p)^2 + q$ .

Step 1: Halve the x coefficient (8 becomes 4) and place inside the squared bracket: $(x + 4)^2$ .

Step 2: Subtract the square of that number (16) and add the original constant (5): $(x + 4)^2 - 16 + 5$ .

Answer: $(x + 4)^2 - 11$ .

Example 2

Finding the Turning Point

State the turning point of $y = x^2 - 6x + 10$ .

Step 1: Complete the square: $(x - 3)^2 - 9 + 10 = (x - 3)^2 + 1$ .

Step 2: Extract coordinates $(-p, q)$ . Answer: The turning point is (3, 1).

Example 3

Odd Coefficient (fractions)

Express $x^2 + 5x + 2$ in the form $(x + p)^2 + q$ .

Step 1: Half of 5 is $\tfrac{5}{2}$ — a fraction, which is fine. Write the bracket: $\left(x + \tfrac{5}{2}\right)^2$ .

Step 2: Subtract $\left(\tfrac{5}{2}\right)^2 = \tfrac{25}{4}$ and add the original constant 2:

\left(x + \tfrac{5}{2}\right)^2 - \tfrac{25}{4} + 2

Step 3: Combine the constants — $2 = \tfrac{8}{4}$ , so $-\tfrac{25}{4} + \tfrac{8}{4} = -\tfrac{17}{4}$ :

Answer: $\left(x + \tfrac{5}{2}\right)^2 - \tfrac{17}{4}$ .

Ready to practise?

12 Completing the Square questions with answers and video solutions