Completing the Square Worksheets

Printable “Quadratic Equations” worksheets:
Solve Quadratic Equation (use factoring)
Solve Quadratic Equation (use quadratic formula)
Solve Word Problems using Quadratic Equations
Sketching Quadratic Graphs
Completing the Square

Examples, solutions, videos, and worksheets to help Algebra 1 students learn how to rewrite quadratic expressions given in standard form, ax² + bx + c in the equivalent completed-square form, [a(x–h)]² + k.

How to rewrite expressions in completed-square form?

Completing the square is a method used to rewrite a quadratic expression in a form that reveals its minimum or maximum point, which is helpful for solving equations and graphing.

Steps to Complete the Square When 𝑎 = 1

1. Start with the quadratic expression in the form: 𝑥² + 𝑏𝑥 + 𝑐
   
2. Take half of the coefficient of 𝑥, square it to get (𝑏/2)². Add and subtract (𝑏/2)² in the expression.
   
3. Rewrite the expression as a perfect square.

Steps to Complete the Square When 𝑎 ≠ 1
When the coefficient of 𝑥² is not equal to 1, you’ll need to factor 𝑎 out before completing the square.

1. Start with the quadratic expression in the form: a𝑥² + 𝑏𝑥 + 𝑐
   
2. Factor out 𝑎
   a(x² + b/a x) + c
   
3. Take half of the coefficient of 𝑥, square it to get (𝑏/(2a))². Add and subtract (𝑏/(2a))² in the expression.
   
4. Complete the square inside the parentheses:

• Take half of the coefficient of 𝑥, square it, and add and subtract this squared term inside the parentheses.
• This gives: 𝑎(𝑥² + 𝑏/𝑎 𝑥 + (𝑏/(2𝑎))² − (𝑏/(2𝑎))² ) + c

4. Rewrite as a perfect square and simplify:

• Rewrite the expression in completed square form, adjusting any constants outside the square.

The following example shows how to rewrite an expression in completed-square form.

Have a look at this video if you need to review how to complete the square.

• Completing the Square

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More Completing the Square Worksheets

Printable
(Answers on the second page.)
Completing the Square Worksheet #1 (a = 1)
Completing the Square Worksheet #2 (a ≠ 1)

Related Lessons & Worksheets

Completing the Squares

More Printable Worksheets

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