Worksheets · Algebra 1

Solving quadratics by completing the square worksheet for Algebra 1

The method that works on every quadratic and explains where the formula comes from. Easy gives whole answers; the harder levels leave a root in, which is the exact answer a test wants.

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

Solving by Completing the Square

Date Period

Solve each equation by completing the square.

  1. Solve by completing the square

    a2−16a+25=0
  2. 2x2+4x−9=0
  3. y2+18y+47=0
  4. x2+7x−9=0
  5. y2−14y+28=0
  6. x2−x−5=0
  7. v2+16v+43=0
  8. x2−7x−12=0
  9. v2+14v+41=0
  10. 2x2−8x+2=0
  11. p2−14p+26=0
  12. 2x2−6x−5=0
  13. a2+16a+62=0
  14. 2x2+9x+1=0
  15. b2−16b+44=0
  16. 2x2−x−7=0
  17. m2+16m+35=0
  18. x2+x−1=0
  19. b2−16b+29=0
  20. x2+7x−5=0

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

Sigma Prep · sigmaprep.io/worksheets

Name

Solving by Completing the SquareAnswer keyVersion 1

Date Period

Solve each equation by completing the square.

  1. Solve by completing the square

    a2−16a+25=0{8−39,8+39}
  2. 2x2+4x−9=0{−2−222,−2+222}
  3. y2+18y+47=0{−9−34,−9+34}
  4. x2+7x−9=0{−7−852,−7+852}
  5. y2−14y+28=0{7−21,7+21}
  6. x2−x−5=0{1−212,1+212}
  7. v2+16v+43=0{−8−21,−8+21}
  8. x2−7x−12=0{7−972,7+972}
  9. v2+14v+41=0{−7−22,−7+22}
  10. 2x2−8x+2=0{2−3,2+3}
  11. p2−14p+26=0{7−23,7+23}
  12. 2x2−6x−5=0{3−192,3+192}
  13. a2+16a+62=0{−8−2,−8+2}
  14. 2x2+9x+1=0{−9−734,−9+734}
  15. b2−16b+44=0{8−25,8+25}
  16. 2x2−x−7=0{1−574,1+574}
  17. m2+16m+35=0{−8−29,−8+29}
  18. x2+x−1=0{−1−52,−1+52}
  19. b2−16b+29=0{8−35,8+35}
  20. x2+7x−5=0{−7−692,−7+692}

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

How to do these

x2+6x−7=0

  1. Move the 7 across: x² + 6x = 7.
  2. Half of 6 is 3, and 3² is 9. Add 9 to both sides.
  3. (x + 3)² = 16, so x + 3 = ±4.

The answer is {1,−7}.

Where students go wrong

Adding the number to one side only. Completing the square adds something to the left, so the same thing has to go on the right or the equation stops being true. The other one is losing the negative root when you take the square root.

Also called completing the square to solve or solving quadratics by the square root method.

What you can put on this worksheet

Solve by completing the square
v2+10v+24=0 → {−6,−4}
Odd b, or a leading coefficient
x2−7x+6=0 → {1,6}
Rearrange first, irrational roots
Solve by completing the square: k2+10k+7=−16 → {−5−2,−5+2}
Starting from x^2 + bx = c, odd b
Solve by completing the square: n2+8n=58 → {−4−74,−4+74}

Questions about these worksheets

Yes. Every solving quadratics by 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 1 is usually taken around 8th or 9th 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 solving quadratics by 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 4: solve by completing the square, odd b, or a leading coefficient, rearrange first, irrational roots and starting from x^2 + bx = c, odd b. Tick as many as you want and set how many of each, or let it spread them evenly.

Adding the number to one side only. Completing the square adds something to the left, so the same thing has to go on the right or the equation stops being true. The other one is losing the negative root when you take the square root.