How to Solve a Quadratic Equation by Completing the Square?

Completing the square is a powerful algebraic technique for solving quadratic equations that also reveals the vertex of the parabola. It works on every quadratic equation — whether or not it factors — and is the method used to derive the quadratic formula itself. Once you learn the steps, completing the square becomes a reliable tool in your Algebra 1 toolkit.

What Is Completing the Square?

Completing the square means rewriting a quadratic expression \(\color{blue}{\text{ ax }^{2} + \text{ bx } + c}\) as a perfect square trinomial plus (or minus) a constant. The result is the vertex form \(\color{blue}{a(x – h)^{2} + k}\), from which the solutions follow immediately by taking square roots.

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How to Complete the Square

Step 1: Ensure the Leading Coefficient Is 1

If \(\color{blue}{a \ne 1}\), divide the entire equation by \(\color{blue}{a}\) first.

• \(\color{blue}{2x^{2} + 8x – 10 = 0}\)  →  divide by 2: \(\color{blue}{x^{2} + 4x – 5 = 0}\)

Step 2: Move the Constant to the Right Side

• \(\color{blue}{x^{2} + 4x = 5}\)

Step 3: Add (\(\color{blue}{\frac{b}{2}}\))² to Both Sides

Take half the coefficient of \(\color{blue}{x}\), square it, and add to both sides. This creates a perfect square trinomial on the left.

• Half of \(\color{blue}{4 = 2}\); \(\color{blue}{2^{2} = 4}\). Add 4 to both sides: \(\color{blue}{x^{2} + 4x + 4 = 9}\)

Step 4: Write the Left Side as a Squared Binomial

• \(\color{blue}{(x + 2)^{2} = 9}\)

Step 5: Take the Square Root of Both Sides

• \(\color{blue}{x + 2 = \pm 3}\)  →  \(\color{blue}{x = 1}\) or \(\color{blue}{x = -5}\)

Step-by-Step Summary

1. If \(\color{blue}{a \ne 1}\), divide the whole equation by \(\color{blue}{a}\).
2. Move the constant term to the right side.
3. Add \(\color{blue}{(\frac{b}{2})^{2}}\) to both sides.
4. Factor the left side as \(\color{blue}{(x + \frac{b}{2})^{2}}\).
5. Take the square root of both sides (include ±).
6. Solve for \(\color{blue}{x}\).

Watch: Solving Quadratic Equations by Completing the Square

Khan Academy demonstrates the full completing-the-square process:

Completing the Square – Worked Examples

Example 1: Solve \(\color{blue}{x^{2} + 6x + 1 = 0}\).

Move constant: \(\color{blue}{x^{2} + 6x = -1}\).
Half of \(\color{blue}{6 = 3}\); add \(\color{blue}{3^{2} = 9}\): \(\color{blue}{x^{2} + 6x + 9 = 8}\).
Factor: \(\color{blue}{(x + 3)^{2} = 8}\).
Take square root: \(\color{blue}{x + 3 = \pm 2\sqrt{2}}\).
Solutions: \(\color{blue}{x = -3 + 2\sqrt{2} \approx -0.17}\) and \(\color{blue}{x = -3 – 2\sqrt{2} \approx -5.83}\).

Example 2: Solve \(\color{blue}{x^{2} – 8x + 7 = 0}\).

Move constant: \(\color{blue}{x^{2} – 8x = -7}\).
Half \(\color{blue}{\text{ of } -8}\) = −4; add \(\color{blue}{16}\): \(\color{blue}{x^{2} – 8x + 16 = 9}\).
Factor: \(\color{blue}{(x – 4)^{2} = 9}\).
Take square root: \(\color{blue}{x – 4 = \pm 3}\).
Solutions: \(\color{blue}{x = 7}\) or \(\color{blue}{x = 1}\). (Check: \(\color{blue}{49-56+7=0}\) ✓; \(\color{blue}{1-8+7=0}\) ✓)

Example 3: Solve \(\color{blue}{x^{2} + 4x – 5 = 0}\) by completing the square.

Move constant: \(\color{blue}{x^{2} + 4x = 5}\).
Add \(\color{blue}{4}\): \(\color{blue}{(x + 2)^{2} = 9}\).
\(\color{blue}{x + 2 = \pm 3}\)  →  \(\color{blue}{x = 1}\) or \(\color{blue}{x = -5}\).

Example 4: Solve \(\color{blue}{2x^{2} + 12x + 4 = 0}\).

Divide by 2: \(\color{blue}{x^{2} + 6x + 2 = 0}\).
Move constant: \(\color{blue}{x^{2} + 6x = -2}\).
Add 9: \(\color{blue}{(x + 3)^{2} = 7}\).
Solutions: \(\color{blue}{x = -3 \pm \sqrt{7}}\).

More Practice: Completing the Square Worked Example

Khan Academy solves \(\color{blue}{x^{2} – 2x – 8 = 0}\) step by step using this method:

Exercises: Completing the Square

1. Solve: \(\color{blue}{x^{2} + 10x + 9 = 0}\)
2. Solve: \(\color{blue}{x^{2} – 6x + 5 = 0}\)
3. Solve: \(\color{blue}{x^{2} + 2x – 3 = 0}\)
4. Solve: \(\color{blue}{x^{2} – 4x – 1 = 0}\)
5. Solve: \(\color{blue}{2x^{2} + 8x – 10 = 0}\)

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Answers

1. \(\color{blue}{x = -1}\) and \(\color{blue}{x = -9}\)
2. \(\color{blue}{x = 1}\) and \(\color{blue}{x = 5}\)
3. \(\color{blue}{x = 1}\) and \(\color{blue}{x = -3}\)
4. \(\color{blue}{x = 2 \pm \sqrt{5}}\)  ≈  \(\color{blue}{4.24}\) or \(\color{blue}{-0.24}\)
5. \(\color{blue}{x = 1}\) and \(\color{blue}{x = -5}\)  (divide by 2 first: \(\color{blue}{x^{2} + 4x – 5 = 0}\))

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Free Solving a Quadratic Equation by Completing the Square Worksheet

Ready to practice on your own? Download our free Solving a Quadratic Equation by Completing the Square worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again — steady practice is the surest way to master Solving a Quadratic Equation by Completing the Square before a quiz or test.

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Frequently Asked Questions

Why learn completing the square when the quadratic formula exists?

Completing the square is how the quadratic formula is derived. It also gives you the vertex form of the parabola directly, which is useful for graphing. Some problems in calculus and advanced algebra also require this technique.

What if the coefficient of x² is not 1?

Divide the entire equation by the leading coefficient before starting. This ensures the perfect-square pattern works cleanly.

How do I know what number to add to complete the square?

Take half the coefficient of \(\color{blue}{x}\), then square it. Always add this to both sides of the equation to keep it balanced.

Related Topics

• Solving a Quadratic Equation
• Factoring Quadratics
• Discriminant of a Quadratic Equation
• Quadratic Function
• Graphing Quadratic Functions