Algebra · Step-by-step guide

Completing the Square

Complete the square to solve quadratic equations and convert quadratics to vertex form.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

For x2+bx\displaystyle x^2+bx, add (b/2)2\displaystyle (b/2)^2 to create (x+b/2)2\displaystyle (x+b/2)^2. In an equation, add the same value to both sides; if the x² coefficient is not 1, divide or factor it first.

x2+bx+(b2)2=(x+b2)2\displaystyle x^2+bx+\left(\frac b2\right)^2=\left(x+\frac b2\right)^2
01

How it works

Build the method from meaning before memorizing the moves.

These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.

1

Make x² monic

The shortcut uses the coefficient of x only after the x² coefficient equals 1.

2

Half, then square

Half the x-coefficient and square that number. It becomes both the added constant and the binomial constant.

3

Maintain balance

When solving, whatever is added on one side must be added on the other.

02

See the structure

A picture makes the relationships easier to remember.

−22vertex (0, −1)
The coordinate graph marks the vertex and the two x-intercepts of a representative quadratic.
03

Worked examples

Follow the reason for each line, then try to reproduce it without looking.

Example 1Create a square

Complete x2+10x\displaystyle x^2+10x.

  1. Half 10 to get 5.
  2. Square 5 to get 25.
  3. Add 25 and factor.
Answerx2+10x+25=(x+5)2\displaystyle x^2+10x+25=(x+5)^2
Example 2Solve

Solve x2+6x−7=0\displaystyle x^2+6x-7=0.

  1. Move −7: x2+6x=7\displaystyle x^2+6x=7.
  2. Add 9 to both sides: (x+3)2=16\displaystyle (x+3)^2=16.
  3. Take both square roots: x+3=±4\displaystyle x+3=\pm4.
Answerx=1,−7\displaystyle x=1,-7
Example 3Vertex form

Rewrite y=x2−8x+3\displaystyle y=x^2-8x+3.

  1. Group x2−8x\displaystyle x^2-8x.
  2. Add and subtract 16.
  3. Factor the perfect square and combine constants.
Answery=(x−4)2−13\displaystyle y=(x-4)^2-13
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Using b²/2 instead of (b/2)².

  • Adding a value to only one side of an equation.

  • Forgetting ± after taking a square root.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Completing the Square

Practice perfect-square construction, solving, and vertex form.

Preview the worksheet

A strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.

06

Practice problems with worked solutions

Solve each problem on paper, use the hint only if needed, then compare every step.

1

Complete the square: x2+8x+3\displaystyle x^2+8x+3.

Show hint

Add and subtract (8/2)2\displaystyle (8/2)^2.

Show worked solution
  1. Half of 8 is 4; square it to get 16.
  2. x2+8x+16−16+3\displaystyle x^2+8x+16-16+3.
  3. Combine constants.

Answer: (x+4)2−13\displaystyle (x+4)^2-13

2

Solve x2+6x−7=0\displaystyle x^2+6x-7=0 by completing the square.

Show hint

Move the constant first.

Show worked solution
  1. x2+6x=7\displaystyle x^2+6x=7.
  2. Add 9: (x+3)2=16\displaystyle (x+3)^2=16.
  3. x+3=±4\displaystyle x+3=\pm4.

Answer: x=1\displaystyle x=1 or x=−7\displaystyle x=-7.

3

Write x2−10x+1\displaystyle x^2-10x+1 in vertex form.

Show hint

Half −10, then square.

Show worked solution
  1. Half is −5 and its square is 25.
  2. x2−10x+25−25+1\displaystyle x^2-10x+25-25+1.
  3. Simplify.

Answer: (x−5)2−24\displaystyle (x-5)^2-24

4

Solve x2−4x+8=0\displaystyle x^2-4x+8=0.

Show hint

Complete the square and interpret the negative result.

Show worked solution
  1. x2−4x=−8\displaystyle x^2-4x=-8.
  2. Add 4: (x−2)2=−4\displaystyle (x-2)^2=-4.
  3. Take square roots using i\displaystyle i.

Answer: x=2±2i\displaystyle x=2\pm2i

5

Write 2x2+12x+5\displaystyle 2x^2+12x+5 in vertex form.

Show hint

Factor 2 from the x-terms first.

Show worked solution
  1. 2(x2+6x)+5\displaystyle 2(x^2+6x)+5.
  2. Add and subtract 9 inside: 2((x+3)2−9)+5\displaystyle 2((x+3)^2-9)+5.
  3. Simplify constants.

Answer: 2(x+3)2−13\displaystyle 2(x+3)^2-13

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

For x2+bx\displaystyle x^2+bx, add (b/2)2\displaystyle (b/2)^2 to create (x+b/2)2\displaystyle (x+b/2)^2. In an equation, add the same value to both sides; if the x² coefficient is not 1, divide or factor it first.

How do I know whether I understand this topic?

You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.

What should I do if I keep making the same mistake?

Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.

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