Solve by completing the square: 4k^2 - 5k + 13 = 0

5 + isqrt(183)8, 5 - isqrt(183)8

Completing the square worksheet for Algebra 2

Completing the square turns a quadratic into a squared parenthesis plus a leftover, which is what makes the vertex readable straight off the page. The same step finds the c that makes a perfect square trinomial, and solves quadratic equations, including ones whose roots turn out complex.

Also taught in Algebra 1, at its own level and with its own examples.

Not quite the one you need? Same words, different skill:

Solving Quadratics by Factoring (Algebra 2). Factor, then set each factor to zero.
Solving Quadratics by Square Roots (Algebra 2). Isolate the square, then take the root. Both of them.
Solving by Completing the Square (Algebra 2). Complete the square, then take the root.
Quadratics with Complex Solutions (Algebra 2). When the discriminant is negative, the answers are complex.
Square Roots (Pre-Algebra). Perfect squares, and estimating the ones between.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Completing the Square Date Period Complete the square as each question asks. Write in vertex form Write in vertex form: $y = x^{2} + 20 x + 113$ . Write in vertex form: $y = - x^{2} + 6 x + 6$ Find the vertex form of $y = x^{2} - 3 x$ . Write in vertex form: $y = - x^{2} + 10 x - 35$ Complete the square: $y = x^{2} + 7 x + 14$ . Write in vertex form: $y = - x^{2} - 10 x - 11$ Put $y = x^{2} - 16 x + 52$ in vertex form. Write in vertex form: $y = 3 x^{2} - 12 x + 21$ Rewrite $y = x^{2} - 18 x + 101$ in the form $y = {(x - h)}^{2} + k$ . Write in vertex form: $y = - 2 x^{2} + 8 x - 11$ Find $c$ to complete the square Find $c$ so that $x^{2} - 8 x + c$ is a perfect square trinomial Find $c$ so that $x^{2} + 7 x + c$ is a perfect square trinomial Find $c$ so that $x^{2} + x + c$ is a perfect square trinomial Find $c$ so that $x^{2} - 12 x + c$ is a perfect square trinomial Find $c$ so that $x^{2} + 13 x + c$ is a perfect square trinomial Solve by completing the square $n^{2} - 14 n + 29 = 0$ $r^{2} + 14 r + 42 = 0$ $v^{2} + 14 v + 10 = 0$ $k^{2} - 14 k + 19 = 0$ $r^{2} - 14 r + 23 = 0$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Completing the Square

Date Period

Complete the square as each question asks.

Write in vertex form
Write in vertex form: $y = x^{2} + 20 x + 113$ .
Write in vertex form: $y = - x^{2} + 6 x + 6$
Find the vertex form of $y = x^{2} - 3 x$ .
Write in vertex form: $y = - x^{2} + 10 x - 35$
Complete the square: $y = x^{2} + 7 x + 14$ .
Write in vertex form: $y = - x^{2} - 10 x - 11$
Put $y = x^{2} - 16 x + 52$ in vertex form.
Write in vertex form: $y = 3 x^{2} - 12 x + 21$
Rewrite $y = x^{2} - 18 x + 101$ in the form $y = {(x - h)}^{2} + k$ .
Write in vertex form: $y = - 2 x^{2} + 8 x - 11$

Find $c$ to complete the square
Find $c$ so that $x^{2} - 8 x + c$ is a perfect square trinomial
Find $c$ so that $x^{2} + 7 x + c$ is a perfect square trinomial
Find $c$ so that $x^{2} + x + c$ is a perfect square trinomial
Find $c$ so that $x^{2} - 12 x + c$ is a perfect square trinomial
Find $c$ so that $x^{2} + 13 x + c$ is a perfect square trinomial

Solve by completing the square
$n^{2} - 14 n + 29 = 0$
$r^{2} + 14 r + 42 = 0$
$v^{2} + 14 v + 10 = 0$
$k^{2} - 14 k + 19 = 0$
$r^{2} - 14 r + 23 = 0$



Sigma Prep · sigmaprep.io/worksheets Name Completing the SquareAnswer keyVersion 1 Date Period Complete the square as each question asks. Write in vertex form Write in vertex form: $y = x^{2} + 20 x + 113$ . $y = {(x + 10)}^{2} + 13$ Write in vertex form: $y = - x^{2} + 6 x + 6$ $y = - {(x - 3)}^{2} + 15$ Find the vertex form of $y = x^{2} - 3 x$ . $y = {(x - \frac{3}{2})}^{2} - \frac{9}{4}$ Write in vertex form: $y = - x^{2} + 10 x - 35$ $y = - {(x - 5)}^{2} - 10$ Complete the square: $y = x^{2} + 7 x + 14$ . $y = {(x + \frac{7}{2})}^{2} + \frac{7}{4}$ Write in vertex form: $y = - x^{2} - 10 x - 11$ $y = - {(x + 5)}^{2} + 14$ Put $y = x^{2} - 16 x + 52$ in vertex form. $y = {(x - 8)}^{2} - 12$ Write in vertex form: $y = 3 x^{2} - 12 x + 21$ $y = 3 {(x - 2)}^{2} + 9$ Rewrite $y = x^{2} - 18 x + 101$ in the form $y = {(x - h)}^{2} + k$ . $y = {(x - 9)}^{2} + 20$ Write in vertex form: $y = - 2 x^{2} + 8 x - 11$ $y = - 2 {(x - 2)}^{2} - 3$ Find $c$ to complete the square Find $c$ so that $x^{2} - 8 x + c$ is a perfect square trinomial $16$ Find $c$ so that $x^{2} + 7 x + c$ is a perfect square trinomial $\frac{49}{4}$ Find $c$ so that $x^{2} + x + c$ is a perfect square trinomial $\frac{1}{4}$ Find $c$ so that $x^{2} - 12 x + c$ is a perfect square trinomial $36$ Find $c$ so that $x^{2} + 13 x + c$ is a perfect square trinomial $\frac{169}{4}$ Solve by completing the square $n^{2} - 14 n + 29 = 0$ ${7 - 2 \sqrt{5}, 7 + 2 \sqrt{5}}$ $r^{2} + 14 r + 42 = 0$ ${- 7 - \sqrt{7}, - 7 + \sqrt{7}}$ $v^{2} + 14 v + 10 = 0$ ${- 7 - \sqrt{39}, - 7 + \sqrt{39}}$ $k^{2} - 14 k + 19 = 0$ ${7 - \sqrt{30}, 7 + \sqrt{30}}$ $r^{2} - 14 r + 23 = 0$ ${7 - \sqrt{26}, 7 + \sqrt{26}}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Completing the SquareAnswer keyVersion 1

Date Period

Complete the square as each question asks.

Write in vertex form
Write in vertex form: $y = x^{2} + 20 x + 113$ . $y = {(x + 10)}^{2} + 13$
Write in vertex form: $y = - x^{2} + 6 x + 6$ $y = - {(x - 3)}^{2} + 15$
Find the vertex form of $y = x^{2} - 3 x$ . $y = {(x - \frac{3}{2})}^{2} - \frac{9}{4}$
Write in vertex form: $y = - x^{2} + 10 x - 35$ $y = - {(x - 5)}^{2} - 10$
Complete the square: $y = x^{2} + 7 x + 14$ . $y = {(x + \frac{7}{2})}^{2} + \frac{7}{4}$
Write in vertex form: $y = - x^{2} - 10 x - 11$ $y = - {(x + 5)}^{2} + 14$
Put $y = x^{2} - 16 x + 52$ in vertex form. $y = {(x - 8)}^{2} - 12$
Write in vertex form: $y = 3 x^{2} - 12 x + 21$ $y = 3 {(x - 2)}^{2} + 9$
Rewrite $y = x^{2} - 18 x + 101$ in the form $y = {(x - h)}^{2} + k$ . $y = {(x - 9)}^{2} + 20$
Write in vertex form: $y = - 2 x^{2} + 8 x - 11$ $y = - 2 {(x - 2)}^{2} - 3$

Find $c$ to complete the square
Find $c$ so that $x^{2} - 8 x + c$ is a perfect square trinomial $16$
Find $c$ so that $x^{2} + 7 x + c$ is a perfect square trinomial $\frac{49}{4}$
Find $c$ so that $x^{2} + x + c$ is a perfect square trinomial $\frac{1}{4}$
Find $c$ so that $x^{2} - 12 x + c$ is a perfect square trinomial $36$
Find $c$ so that $x^{2} + 13 x + c$ is a perfect square trinomial $\frac{169}{4}$

Solve by completing the square
$n^{2} - 14 n + 29 = 0$ ${7 - 2 \sqrt{5}, 7 + 2 \sqrt{5}}$
$r^{2} + 14 r + 42 = 0$ ${- 7 - \sqrt{7}, - 7 + \sqrt{7}}$
$v^{2} + 14 v + 10 = 0$ ${- 7 - \sqrt{39}, - 7 + \sqrt{39}}$
$k^{2} - 14 k + 19 = 0$ ${7 - \sqrt{30}, 7 + \sqrt{30}}$
$r^{2} - 14 r + 23 = 0$ ${7 - \sqrt{26}, 7 + \sqrt{26}}$

How to do these

$y = x^{2} + 6 x + 2$

Halve the middle coefficient: 6 halves to 3.
(x + 3)² is x² + 6x + 9, which is 7 too big.
So subtract 7.

The answer is $y = {(x + 3)}^{2} - 7$ .

Where students go wrong

Halving but not squaring, or squaring but not subtracting. Both steps are needed: half the middle number goes inside the parentheses, and its square has to come back off the outside.

Also called completing the square method or making a perfect square.

What you can put on this worksheet

Rewrite by completing the square: Find the vertex form of $y = x^{2} + 10 x + 20$ . → $y = {(x + 5)}^{2} - 5$
Leading coefficient other than 1: Write in vertex form: $y = 3 x^{2} - 30 x + 62$ → $y = 3 {(x - 5)}^{2} - 13$
The c that completes the square: Find $c$ so that $x^{2} + 14 x + c$ is a perfect square trinomial → $49$
Solve by completing the square: $v^{2} + 10 v + 24 = 0$ → ${- 6, - 4}$
Odd b, or a leading coefficient: $x^{2} - 7 x + 6 = 0$ → ${1, 6}$
Starting from x^2 + bx = c, odd b: Solve by completing the square: $n^{2} + 8 n = 58$ → ${- 4 - \sqrt{74}, - 4 + \sqrt{74}}$
By completing the square: Solve by completing the square: $v^{2} - 8 v + 17 = 0$ → ${4 + i, 4 - i}$
Fraction b, a not 1 with complex roots: Solve by completing the square: $4 k^{2} - 5 k + 13 = 0$ → ${\frac{5 + i \sqrt{183}}{8}, \frac{5 - i \sqrt{183}}{8}}$

Questions about these worksheets

Yes. Every completing the square sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Algebra 2 is usually taken around 10th or 11th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the completing the square questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 8: rewrite by completing the square, leading coefficient other than 1, the c that completes the square, solve by completing the square, odd b, or a leading coefficient, starting from x^2 + bx = c, odd b, by completing the square and fraction b, a not 1 with complex roots. Tick as many as you want and set how many of each, or let it spread them evenly.

Halving but not squaring, or squaring but not subtracting. Both steps are needed: half the middle number goes inside the parentheses, and its square has to come back off the outside.

Completing the square worksheet for Algebra 2

Select your difficulty

Write in vertex form

Find ccc to complete the square

Solve by completing the square

Write in vertex form

Find ccc to complete the square

Solve by completing the square

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these completing the square worksheets free?+

Do the completing the square worksheets come with an answer key?+

Are these completing the square worksheets printable?+

What grade are these completing the square worksheets for?+

Can I get a different version of the same completing the square worksheet?+

Is this a free alternative to Kuta Software for completing the square?+

Can I choose how hard the completing the square questions are?+

What kinds of question are on a completing the square worksheet?+

What do students get wrong with completing the square?+

Find $c$ to complete the square

Find $c$ to complete the square