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Lesson overview

Algebra - quadratic expressions

Completing the square is a powerful algebraic technique that rewrites a quadratic expression into a form that reveals the vertex of its parabola. For a quadratic $x^2 + bx + c$ , we can rewrite it as $(x + \frac{b}{2})^2 - \frac{b^2}{4} + c$ . This form makes it easy to find the minimum or maximum value (the vertex), solve equations without using the formula, and understand the transformations of parabolic graphs. For quadratics with a leading coefficient $a \neq 1$ , we factor out $a$ first, complete the square for the expression in brackets, then distribute.

Completing the square for $x^2 + 6x + 2$

Rewrite $x^2 + 6x + 2$ in the form $(x + p)^2 + q$ .

Half the coefficient of $x$ is $\frac{6}{2} = 3$ . Square it: $3^2 = 9$ . Subtract the squared term and simplify.

Finding the vertex from completed square form

Find the vertex of the parabola $y = (x - 2)^2 + 5$ .

Form $y = (x - 2)^2 + 5$

$(x - 2)^2$ is zero when $x = 2$

Minimum value $y = 5$

Occurs at $x = 2$

Vertex $(2, 5)$

The turning point of the parabola

The vertex is $(−p, q)$ from the form $(x + p)^2 + q$ . Here, the form is $(x - 2)^2 + 5$ , so the vertex is $(2, 5)$ .

Completing the square when $a \neq 1$

Write $2x^2 + 8x + 3$ in the form $a(x + p)^2 + q$ .

Factor out the coefficient of $x^2$ first. Complete the square inside the brackets. Then distribute and simplify.

Using completed square form to solve equations

Solve $x^2 + 4x - 5 = 0$ using completing the square.

After completing the square, take square roots of both sides, remembering both positive and negative roots.

Worked example: Complete the square and find the minimum

For the quadratic $f(x) = 3x^2 - 12x + 7$ , complete the square and find the minimum value.

Factor out 3 from the first two terms: $3(x^2 - 4x) + 7$ .
To complete the square inside brackets, take half of the coefficient of $x$ : $\frac{-4}{2} = -2$ .
Square it: $(-2)^2 = 4$ . Write as: $3((x - 2)^2 - 4) + 7$ .
Distribute the 3: $3(x - 2)^2 - 12 + 7$ .
Simplify: $3(x - 2)^2 - 5$ .
The minimum value is $-5$ (when $(x - 2)^2 = 0$ , i.e., when $x = 2$ ).

Try it

Read each question carefully. Complete the square and identify the vertex or solve the equation. Choose the correct answer from the four options.

Question 1

Write $x^2 + 8x + 3$ in the form $(x + p)^2 + q$ .

💡 Half of 8 is 4. Square it to get 16. So $x^2 + 8x + 3 = (x + 4)^2 - 16 + 3 = (x + 4)^2 - 13$ .

Question 2

Find the vertex of the parabola $y = (x + 3)^2 - 7$ .

💡 In the form $(x + p)^2 + q$ , the vertex is $(-p, q)$ . Here, $p = 3$ and $q = -7$ , so the vertex is $(-3, -7)$ .

Question 3

Complete the square: $2x^2 - 12x + 5 = 2(x - p)^2 + q$ . What is $q$ ?

💡 Factor: $2(x^2 - 6x) + 5$ . Complete: $2((x - 3)^2 - 9) + 5 = 2(x - 3)^2 - 18 + 5 = 2(x - 3)^2 - 13$ .

Question 4

Solve $x^2 + 6x + 8 = 0$ by completing the square.

💡 $(x + 3)^2 - 9 + 8 = 0 \Rightarrow (x + 3)^2 = 1 \Rightarrow x + 3 = \pm 1 \Rightarrow x = -2$ or $x = -4$ .

Question 5

What is the minimum value of $f(x) = x^2 - 10x + 21$ ?

💡 Complete the square: $(x - 5)^2 - 25 + 21 = (x - 5)^2 - 4$ . The minimum is $-4$ when $(x - 5)^2 = 0$ .

Common mistakes

Watch for these when working through the lesson.

Forgetting to subtract the squared term after rewriting. If you complete the square to $(x + b)^2$ , you must subtract $b^2$ to maintain equality.
When $a \neq 1$ , failing to factor out $a$ before completing the square, or forgetting to distribute $a$ back through the adjusted constant.
Confusing the sign of $p$ in the vertex. From $(x + p)^2 + q$ , the vertex is $(-p, q)$ . The $x$ -coordinate is the opposite sign.

Completing the square

Lesson overview

Worked example: Complete the square and find the minimum

Try it

Common mistakes

Related topics

Independent practice will plug in here