Completing the Square Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Completing the Square.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A technique for rewriting ax^2 + bx + c in vertex form a(x - h)^2 + k by adding and subtracting the value \left(\frac{b}{2a}\right)^2 to create a perfect square trinomial.

Imagine you have x^2 + 6x and want a perfect square. A perfect square like (x + 3)^2 = x^2 + 6x + 9 needs that extra +9. So you add 9 and subtract 9 to keep the expression equal—then group the perfect square part.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: By strategically adding and subtracting the same value, you reshape an expression into a perfect square plus a constant.

Common stuck point: When a \neq 1, you must factor out a from the x^2 and x terms first, then complete the square inside the parentheses—and be careful when distributing a back.

Sense of Study hint: Take half of the x-coefficient, square it, then add and subtract that value inside the expression.

Worked Examples

Example 1

medium
Rewrite x^2 + 6x + 2 in vertex form by completing the square.

Solution

  1. 1
    Take half of the coefficient of x: \frac{6}{2} = 3.
  2. 2
    Square it: 3^2 = 9.
  3. 3
    Add and subtract 9: x^2 + 6x + 9 - 9 + 2 = (x + 3)^2 - 7.

Answer

(x + 3)^2 - 7
Completing the square transforms a quadratic into vertex form a(x-h)^2 + k by creating a perfect square trinomial. The vertex is (-3, -7).

Example 2

hard
Solve x^2 - 4x - 5 = 0 by completing the square.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
What number completes the square for x^2 + 10x + \,\_\,?

Example 2

medium
Rewrite x^2 - 8x + 20 in vertex form.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

quadratic standard formexpressions