Worksheets · Precalculus

Quadratic equations with complex solutions worksheet for Precalculus

In Precalculus complex solutions are the non-real zeros a polynomial can have, and they always come in conjugate pairs. These sheets find them in quadratics with a negative discriminant, solving by square roots, by the quadratic formula or by completing the square.

Also taught in Algebra 2, at its own level and with its own examples.

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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.

  1. Solve by taking square roots

    r2+16=0
  2. n2+1=0
  3. r2+100=0
  4. v2+49=0
  5. b2+4=0
  6. v2+121=0
  7. m2+25=0

  1. Solve with the quadratic formula

    b2+b+27=0
  2. p2−2p+26=0
  3. v2+3v+17=0
  4. r2−r+30=0
  5. n2−5n+29=0
  6. x2+x+8=0
  7. a2+8a+29=0

  1. Solve by completing the square

    Solve by completing the square: a2+4a+12=0
  2. Solve by completing the square: p2−4p+5=0
  3. Solve by completing the square: v2+4v+9=0
  4. Solve by completing the square: a2−10a+27=0
  5. Solve by completing the square: b2+4b+11=0
  6. Solve by completing the square: n2−10n+30=0

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Quadratics with Complex SolutionsAnswer keyVersion 2

Date Period

Solve each equation.

  1. Solve by taking square roots

    r2+16=0{4i,−4i}
  2. n2+1=0{i,−i}
  3. r2+100=0{10i,−10i}
  4. v2+49=0{7i,−7i}
  5. b2+4=0{2i,−2i}
  6. v2+121=0{11i,−11i}
  7. m2+25=0{5i,−5i}

  1. Solve with the quadratic formula

    b2+b+27=0{−12+i1072,−12−i1072}
  2. p2−2p+26=0{1+5i,1−5i}
  3. v2+3v+17=0{−32+i592,−32−i592}
  4. r2−r+30=0{12+i1192,12−i1192}
  5. n2−5n+29=0{52+i912,52−i912}
  6. x2+x+8=0{−12+i312,−12−i312}
  7. a2+8a+29=0{−4+i13, −4−i13}

  1. Solve by completing the square

    Solve by completing the square: a2+4a+12=0{−2+2i2, −2−2i2}
  2. Solve by completing the square: p2−4p+5=0{2+i,2−i}
  3. Solve by completing the square: v2+4v+9=0{−2+i5,−2−i5}
  4. Solve by completing the square: a2−10a+27=0{5+i2,5−i2}
  5. Solve by completing the square: b2+4b+11=0{−2+i7,−2−i7}
  6. Solve by completing the square: n2−10n+30=0{5+i5,5−i5}

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

x2−4x+13=0

  1. Discriminant: 16 - 52 = -36, so the zeros are not real.
  2. x = (4 ± √(-36))/2 = (4 ± 6i)/2.

The answer is {2+3i,2−3i}.

Where students go wrong

Dividing only the 6i by 2. Both parts of 4 ± 6i are divided, giving 2 ± 3i.

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

What you can put on this worksheet

By square roots
v2+49=0 → {7i,−7i}
By the quadratic formula
v2+v+22=0 → {−12+i872,−12−i872}
By completing the square
Solve by completing the square: v2−8v+17=0 → {4+i,4−i}
Fraction and imaginary roots, a single root at zero
−34+19m2=−54 → {2i9519,−2i9519}

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 4: by square roots, by the quadratic formula, by completing the square and fraction and imaginary roots, a single root at zero. 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.