Worksheets · Precalculus

Precalculus radical and rational equations worksheet

In Precalculus rational equations come from rational functions, so the restricted values are the domain you already found. Clearing the denominators often leaves a quadratic, and one of its roots can be a value the domain throws out.

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

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.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Radical and Rational Equations

Date Period

Solve each equation.

  1. 2a+122=12
  2. 20n−6=−11
  3. xx−3−2x=9x(x−3)
  4. 4p−39+5=10
  5. −42y+14=7
  6. xx+1−2x=1x(x+1)
  7. 4k+5+9=18
  8. −72a+5=−4
  9. xx+2+7x=4x(x+2)
  10. 5p−19=4
  11. −52m−10=−6
  12. xx−7+2x=49x(x−7)
  13. 3v−20−3=1
  14. 10m+5=6
  15. xx+1−1x=1x(x+1)
  16. 5a−69−7=−3
  17. −14v+1=0
  18. xx+6+14x=36x(x+6)
  19. 3m+40−6=4
  20. −4m−2=0

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

Sigma Prep · sigmaprep.io/worksheets

Name

Radical and Rational EquationsAnswer keyVersion 1

Date Period

Solve each equation.

  1. 2a+122=12{11}
  2. 20n−6=−11{−4}
  3. xx−3−2x=9x(x−3){−1}
  4. 4p−39+5=10{16}
  5. −42y+14=7{6}
  6. xx+1−2x=1x(x+1){3}
  7. 4k+5+9=18{19}
  8. −72a+5=−4{8}
  9. xx+2+7x=4x(x+2){−5}
  10. 5p−19=4{7}
  11. −52m−10=−6{−13}
  12. xx−7+2x=49x(x−7){−9}
  13. 3v−20−3=1{12}
  14. 10m+5=6{10}
  15. xx+1−1x=1x(x+1){2}
  16. 5a−69−7=−3{17}
  17. −14v+1=0{14}
  18. xx+6+14x=36x(x+6){−8}
  19. 3m+40−6=4{20}
  20. −4m−2=0{−2}

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

How to do these

2x−1+xx+1=4x2−1

  1. The common denominator is (x - 1)(x + 1), so x cannot be 1 or -1.
  2. Multiply through: 2(x + 1) + x(x - 1) = 4, which is x² + x - 2 = 0.
  3. Factor: (x + 2)(x - 1) = 0, so x = -2 or x = 1.
  4. x = 1 is excluded, so only -2 is left.

The answer is x=−2.

Where students go wrong

Keeping a root that makes a denominator zero. Write the excluded values down before you multiply, then cross off any answer on that list.

Also called equations with fractions.

What you can put on this worksheet

A square root equation
v+42=7 → {7}
An equation with a variable underneath
30v=5 → {6}
An extraneous root to throw out
xx−3−4x=9x(x−3) → {1}
Quadratic results and extraneous solutions
x−15x=2 → {−3,5}
Squared letters, trinomial and cubic denominators, cross terms
k−52k=22k+3 → {−75}
Quadratics on top
a2−6aa2+2a+2a=a−52a → {35}

Questions about these worksheets

Yes. Every radical and rational equations 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 radical and rational equations 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 6: a square root equation, an equation with a variable underneath, an extraneous root to throw out, quadratic results and extraneous solutions, squared letters, trinomial and cubic denominators, cross terms and quadratics on top. Tick as many as you want and set how many of each, or let it spread them evenly.

Not checking the answer. Squaring both sides can invent solutions that do not work in the original equation, so every answer has to go back in. With a denominator the danger is the opposite: a value that makes the bottom zero has to be thrown out.