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Algebra 2 Completing the Square (Show all work) - Printable Mathematics Worksheets - Wayground

Algebra 2 Completing the Square (Show all work)

10 questions

10th - 12th Grade

Mathematics

This Grade 10–12 mathematics worksheet gives students focused practice completing the square and solving quadratic equations. Across 10 multiple-choice questions, students determine the constant needed to form a perfect-square trinomial, use that form to find roots, identify the updated constant term, and solve equations with different coefficients and presentations. The problems include equations such as x² + 6x − 4 = 36, n² − 2n − 3 = 0, 3x² + 5x − 12 = 0, and (x − 3)² = 49. Although students select from answer choices, the title prompts them to show all work, making the worksheet useful for checking the completing-the-square process rather than answers alone. It is designed for Algebra 2 instruction, review, independent practice, or assessment in Grades 10–12. This free printable worksheet includes a complete answer key for efficient checking.

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Worksheets

Algebra 2 Completing the Square (Show all work)

Total questions: 10

Name
Class
Date
1.
Complete the square.
x2 + 6x + ____
a)
x2 + 6x + 9
b)
x2 + 6x - 36
c)
x2 + 6x + 36
d)
x2 + 6x - 9
2.
Solve the equation by completing the square and then finding the roots. 
x2+ 6x - 4 = 36
a)
x = -7, 7
b)
x = 4, -10
c)
x = -4 +- √7
d)
x = 10, -4
3.
Solve the following equation by completing the square:
n2 - 2n - 3 = 0
a)
{3 and-1}
b)
{4 and -4}
c)
{5 and -3}
d)
{8 and -7}
4.
Solve by completing the square.
x2 +12x = 5
a)
X = 6 + √41 or  6 − √41
b)
X= 35 or 47
c)
X = −6 + √41 or  −6 − √41
d)
X = √35 or √47
5.
Solve
 3x² + 5x - 12 = 0
a)
x = -4/3, x = 3
b)
x = 4/3, x = -3
c)
x = -1, x = 4
d)
x = 1, x = -4
6.

What is one of the solutions to this equation?

(x - 3)2 = 49

a)

7

b)

-7

c)

-4

d)

4

7.

What is the formula of calculate the new C-term

a)

C = b2\frac{b}{2}

b)

C = −b2a\frac{-b}{2a}

c)

C = (b2)2\left(\frac{b}{2}\right)^2

d)

C = (b4)2\left(\frac{b}{4}\right)^2

8.
Complete the Square and solve the equation.
r2 - 4r = 30
a)
2 ± √ 34
b)
± √ 34
c)
 34
d)
± 2
9.
Complete the Square and solve the equation.
a2 - 2a - 35 = 0
a)
1 ± √ 6
b)
-7, 7
c)
-5, 7
d)
35, 37
10.
Complete the Square and solve the equation.
m2 + 12m + 19 = 0
a)
- 6 ± √ - 19
b)
6 ± √ 19
c)
- 23, - 11
d)
- 6 ± √ 17
Answer Key

Algebra 2 Completing the Square (Show all work)

Total questions: 10

1.
a) x2 + 6x + 9
2.
b) x = 4, -10
3.
a) {3 and-1}
4.
c) X = −6 + √41 or  −6 − √41
5.
b) x = 4/3, x = -3
6.
c)

-4

7.
c)

C = (b2)2\left(\frac{b}{2}\right)^2

8.
a) 2 ± √ 34
9.
c) -5, 7
10.
d) - 6 ± √ 17

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FAQs

What does this completing-the-square worksheet help students learn?

This worksheet helps students complete the square and use the resulting perfect-square form to solve quadratic equations. Its 10 multiple-choice items require students to find the added constant, apply the method to equations such as x² + 12x = 5 and r² − 4r = 30, and identify exact roots, including radical solutions. The “show all work” format supports checking each algebraic step even when students select an answer choice.

How can I use this worksheet to teach completing the square?

Model the first item by showing that half the coefficient of x is squared to produce the needed constant, then connect that trinomial to a perfect square. Have students solve the introductory equations before discussing examples with radical roots, such as x² + 12x = 5, and then compare the process with the nonmonic equation 3x² + 5x − 12 = 0. Ask students to justify why both roots must be checked in items such as (x − 3)² = 49.

What mistakes should I watch for when students complete this worksheet?

Watch for students squaring the full x-term coefficient instead of squaring half of it, or changing one side of an equation without making the same change to the other side. Students may also lose the negative sign when rewriting solutions such as −6 ± √41, report only one root, or confuse the constant added to complete the square with the final solution. For the 3x² equation, check that students do not apply the monic procedure without first accounting for the leading coefficient.

How do I assign and grade this completing-the-square worksheet?

The worksheet is available as a printable PDF and as a digital quiz on Wayground, and it includes a complete answer key for all 10 multiple-choice questions. For printable submissions, you can scan student work with the Wayground for Teachers app to support grading while reviewing whether students showed the completing-the-square steps. The printable format can also help schools reduce screen time when appropriate.

How can I differentiate this completing-the-square worksheet for my students?

Use the worksheet’s Advanced Settings to adjust font spacing and font size so the equations and answer choices have more visual room for students to show work. You can also enable a dyslexia-friendly font or translate the worksheet into another language when those options support access to the quadratic-equation practice.

Where can I find more worksheets like this on Wayground?

Wayground offers a comprehensive collection of free printable worksheets and practice problems across subjects, with downloadable PDFs and answer keys to help students build essential skills and concepts. Browse more resources at https://wayground.com/en-us/worksheets.

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