How to Complete the Square: Step-by-Step Guide

What You'll Need

To complete the square by hand, gather these supplies and tools:

• Paper – for writing out the steps.
• Pencil – easy to erase mistakes.
• Eraser – you'll likely need it.
• Scientific calculator – optional, for checking arithmetic or square roots.

If you need a refresher on the definition and purpose, read our What Is Completing the Square? Guide.

Step-by-Step Method

Follow these steps to rewrite any quadratic equation from standard form ax² + bx + c = 0 into vertex form a(x - h)² + k.

1. Write the equation in standard form. Make sure it's ax² + bx + c = 0. If it's not, rearrange terms.
2. If a ≠ 1, factor a out of the x² and x terms. Rewrite as a(x² + (b/a)x) + c = 0. (Leave the constant c outside.)
3. Take half of the coefficient of x (inside the parentheses), square it, and add and subtract that value inside. For x² + (b/a)x, half is (b/(2a)), square is (b/(2a))². So you get a[x² + (b/a)x + (b/(2a))² - (b/(2a))²] + c.
4. Group the perfect square trinomial. The first three terms inside form (x + b/(2a))². Write a[(x + b/(2a))² - (b/(2a))²] + c.
5. Distribute a and combine constants. Multiply a through: a(x + b/(2a))² - a*(b/(2a))² + c. Simplify the constant: k = c - (b²)/(4a). The vertex form is a(x - h)² + k where h = -b/(2a). (Note: the sign inside the parentheses flips: (x + b/(2a)) becomes (x - h) because h = -b/(2a).)

For a deeper look at the formula behind these steps, check out our Completing the Square Formula Derivation.

Worked Example 1: a = 1

Consider x² + 6x + 5 = 0.

1. Standard form: already set.
2. a = 1, so no factoring needed.
3. Coefficient of x is 6. Half is 3, square is 9. Add and subtract inside: x² + 6x + 9 - 9 + 5 = 0.
4. Perfect square: (x + 3)² - 9 + 5 = 0 → (x + 3)² - 4 = 0.
5. Vertex form: (x + 3)² - 4 = 0, so a = 1, h = -3, k = -4. The vertex is (-3, -4).

Worked Example 2: a ≠ 1

Solve 2x² + 8x - 10 = 0.

1. Standard form: fine.
2. a = 2, so factor: 2(x² + 4x) - 10 = 0.
3. Inside x²+4x: half of 4 is 2, square is 4. Add and subtract: 2(x² + 4x + 4 - 4) - 10 = 0.
4. Group: 2[(x + 2)² - 4] - 10 = 0.
5. Distribute: 2(x + 2)² - 8 - 10 = 0 → 2(x + 2)² - 18 = 0. Vertex form: 2(x + 2)² - 18 = 0, so h = -2, k = -18. Vertex = (-2, -18).

After rewriting in vertex form, you can analyze the parabola's properties. Learn how in our Results Interpretation Guide.

Common Pitfalls

• Forgetting to factor out a when a ≠ 1. Many students skip step 2, leading to wrong constants.
• Sign errors when taking half and squaring. Always square after halving, and watch the sign of b.
• Neglecting to multiply the added-and-subtracted square by a. When you distribute a later, the subtracted term is multiplied by a (as we did in example 2).
• Mishandling the constant sign. When moving the constant to the other side, keep it equal. Possibly double-check your arithmetic.
• Confusing vertex form. Remember a(x - h)² + k; the sign inside is opposite of h.

Practice with different coefficients. Use our Completing The Square Calculator to verify your steps.