5/2 + isqrt(67)2, 5/2 - isqrt(67)2

-9/2 + isqrt(7)2, -9/2 - isqrt(7)2

-1/2 + isqrt(87)2, -1/2 - isqrt(87)2

Precalculus quadratic equations with complex solutions worksheet

A quadratic with a negative discriminant has no real solutions, but it has two complex ones, and they are always a pair a + bi and a - bi. The square root of a negative number is i times the square root of the positive one: the square root of -36 is 6i.

Select your difficulty

Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name Quadratics with Complex Solutions Date Period Solve each equation. $a^{2} + 1 = 0$ $v^{2} + 10 v + 30 = 0$ Solve by completing the square: $b^{2} - 8 b + 20 = 0$ $y^{2} + 36 = 0$ $p^{2} + 3 = 0$ Solve by completing the square: $m^{2} - 4 m + 6 = 0$ $x^{2} + 36 = 0$ $v^{2} - 6 v + 20 = 0$ Solve by completing the square: $v^{2} - 2 v + 8 = 0$ $p^{2} + 100 = 0$ $y^{2} - 5 y + 23 = 0$ Solve by completing the square: $n^{2} + 6 n + 12 = 0$ $n^{2} + 81 = 0$ $v^{2} - 2 v + 10 = 0$ Solve by completing the square: $a^{2} - 10 a + 32 = 0$ $y^{2} + 121 = 0$ $y^{2} + 9 y + 22 = 0$ Solve by completing the square: $n^{2} - 10 n + 27 = 0$ $y^{2} + 4 = 0$ $m^{2} - 4 m + 20 = 0$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Quadratics with Complex Solutions

Date Period

Solve each equation.

$a^{2} + 1 = 0$
$v^{2} + 10 v + 30 = 0$
Solve by completing the square: $b^{2} - 8 b + 20 = 0$
$y^{2} + 36 = 0$
$p^{2} + 3 = 0$
Solve by completing the square: $m^{2} - 4 m + 6 = 0$
$x^{2} + 36 = 0$
$v^{2} - 6 v + 20 = 0$
Solve by completing the square: $v^{2} - 2 v + 8 = 0$
$p^{2} + 100 = 0$
$y^{2} - 5 y + 23 = 0$
Solve by completing the square: $n^{2} + 6 n + 12 = 0$
$n^{2} + 81 = 0$
$v^{2} - 2 v + 10 = 0$
Solve by completing the square: $a^{2} - 10 a + 32 = 0$
$y^{2} + 121 = 0$
$y^{2} + 9 y + 22 = 0$
Solve by completing the square: $n^{2} - 10 n + 27 = 0$
$y^{2} + 4 = 0$
$m^{2} - 4 m + 20 = 0$



Sigma Prep · sigmaprep.io/worksheets Name Quadratics with Complex SolutionsAnswer keyVersion 1 Date Period Solve each equation. $a^{2} + 1 = 0$ ${i, - i}$ $v^{2} + 10 v + 30 = 0$ ${- 5 + i \sqrt{5}, - 5 - i \sqrt{5}}$ Solve by completing the square: $b^{2} - 8 b + 20 = 0$ ${4 + 2 i, 4 - 2 i}$ $y^{2} + 36 = 0$ ${6 i, - 6 i}$ $p^{2} + 3 = 0$ ${i \sqrt{3}, - i \sqrt{3}}$ Solve by completing the square: $m^{2} - 4 m + 6 = 0$ ${2 + i \sqrt{2}, 2 - i \sqrt{2}}$ $x^{2} + 36 = 0$ ${6 i, - 6 i}$ $v^{2} - 6 v + 20 = 0$ ${3 + i \sqrt{11}, 3 - i \sqrt{11}}$ Solve by completing the square: $v^{2} - 2 v + 8 = 0$ ${1 + i \sqrt{7}, 1 - i \sqrt{7}}$ $p^{2} + 100 = 0$ ${10 i, - 10 i}$ $y^{2} - 5 y + 23 = 0$ ${\frac{5}{2} + \frac{i \sqrt{67}}{2}, \frac{5}{2} - \frac{i \sqrt{67}}{2}}$ Solve by completing the square: $n^{2} + 6 n + 12 = 0$ ${- 3 + i \sqrt{3}, - 3 - i \sqrt{3}}$ $n^{2} + 81 = 0$ ${9 i, - 9 i}$ $v^{2} - 2 v + 10 = 0$ ${1 + 3 i, 1 - 3 i}$ Solve by completing the square: $a^{2} - 10 a + 32 = 0$ ${5 + i \sqrt{7}, 5 - i \sqrt{7}}$ $y^{2} + 121 = 0$ ${11 i, - 11 i}$ $y^{2} + 9 y + 22 = 0$ ${- \frac{9}{2} + \frac{i \sqrt{7}}{2}, - \frac{9}{2} - \frac{i \sqrt{7}}{2}}$ Solve by completing the square: $n^{2} - 10 n + 27 = 0$ ${5 + i \sqrt{2}, 5 - i \sqrt{2}}$ $y^{2} + 4 = 0$ ${2 i, - 2 i}$ $m^{2} - 4 m + 20 = 0$ ${2 + 4 i, 2 - 4 i}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Quadratics with Complex SolutionsAnswer keyVersion 1

Date Period

Solve each equation.

$a^{2} + 1 = 0$ ${i, - i}$
$v^{2} + 10 v + 30 = 0$ ${- 5 + i \sqrt{5}, - 5 - i \sqrt{5}}$
Solve by completing the square: $b^{2} - 8 b + 20 = 0$ ${4 + 2 i, 4 - 2 i}$
$y^{2} + 36 = 0$ ${6 i, - 6 i}$
$p^{2} + 3 = 0$ ${i \sqrt{3}, - i \sqrt{3}}$
Solve by completing the square: $m^{2} - 4 m + 6 = 0$ ${2 + i \sqrt{2}, 2 - i \sqrt{2}}$
$x^{2} + 36 = 0$ ${6 i, - 6 i}$
$v^{2} - 6 v + 20 = 0$ ${3 + i \sqrt{11}, 3 - i \sqrt{11}}$
Solve by completing the square: $v^{2} - 2 v + 8 = 0$ ${1 + i \sqrt{7}, 1 - i \sqrt{7}}$
$p^{2} + 100 = 0$ ${10 i, - 10 i}$
$y^{2} - 5 y + 23 = 0$ ${\frac{5}{2} + \frac{i \sqrt{67}}{2}, \frac{5}{2} - \frac{i \sqrt{67}}{2}}$
Solve by completing the square: $n^{2} + 6 n + 12 = 0$ ${- 3 + i \sqrt{3}, - 3 - i \sqrt{3}}$
$n^{2} + 81 = 0$ ${9 i, - 9 i}$
$v^{2} - 2 v + 10 = 0$ ${1 + 3 i, 1 - 3 i}$
Solve by completing the square: $a^{2} - 10 a + 32 = 0$ ${5 + i \sqrt{7}, 5 - i \sqrt{7}}$
$y^{2} + 121 = 0$ ${11 i, - 11 i}$
$y^{2} + 9 y + 22 = 0$ ${- \frac{9}{2} + \frac{i \sqrt{7}}{2}, - \frac{9}{2} - \frac{i \sqrt{7}}{2}}$
Solve by completing the square: $n^{2} - 10 n + 27 = 0$ ${5 + i \sqrt{2}, 5 - i \sqrt{2}}$
$y^{2} + 4 = 0$ ${2 i, - 2 i}$
$m^{2} - 4 m + 20 = 0$ ${2 + 4 i, 2 - 4 i}$

How to do these

$x^{2} - 4 x + 13 = 0$

The discriminant is 16 - 52 = -36, so the solutions are complex.
x = (4 plus or minus the square root of -36) / 2 = (4 plus or minus 6i) / 2.

The answer is ${2 + 3 i, 2 - 3 i}$ .

Where students go wrong

Stopping at "no solution". That is right for the real numbers, but once i is available every quadratic has two solutions. The other slip is taking the square root of -36 as -6.

Also called complex solutions, imaginary solutions, quadratic formula complex roots or negative discriminant.

What you can put on this worksheet

By square roots: $v^{2} + 49 = 0$ → ${7 i, - 7 i}$
By the quadratic formula: $v^{2} + v + 22 = 0$ → ${- \frac{1}{2} + \frac{i \sqrt{87}}{2}, - \frac{1}{2} - \frac{i \sqrt{87}}{2}}$
By completing the square: Solve by completing the square: $v^{2} - 8 v + 17 = 0$ → ${4 + i, 4 - i}$

Questions about these worksheets

Yes. Every quadratic equations with complex solutions 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.

Precalculus is usually taken around 11th or 12th 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 quadratic equations with complex solutions 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 3: by square roots, by the quadratic formula and by completing the square. Tick as many as you want and set how many of each, or let it spread them evenly.

Stopping at "no solution". That is right for the real numbers, but once i is available every quadratic has two solutions. The other slip is taking the square root of -36 as -6.

Precalculus quadratic equations with complex solutions worksheet

Select your difficulty

Quadratics with Complex Solutions

Quadratics with Complex SolutionsAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these quadratic equations with complex solutions worksheets free?+

Do the quadratic equations with complex solutions worksheets come with an answer key?+

Are these quadratic equations with complex solutions worksheets printable?+

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Can I get a different version of the same quadratic equations with complex solutions worksheet?+

Is this a free alternative to Kuta Software for quadratic equations with complex solutions?+

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