Completing the Square Worksheets

Printable “Quadratic Equations” worksheets:
Solve Quadratic Equation (use factoring)
Solve Word Problems using Quadratic Equations
Sketching Quadratic Graphs

Rewrite Expressions in Completed-Squared Form
Solve Quadratic Equation (use completing the square)
Solve Quadratic Equation (use quadratic formula)
Discriminant in the Quadratic Formula

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

Click on the following worksheet to get a printable pdf document.
Scroll down the page for more Completing the Square Worksheets.

 

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)

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Completing the Squares

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