Factoring quadratic form worksheet for Precalculus

Precalculus uses quadratic form to factor quartics like x⁴ - 13x² + 36 all the way down and to find their real solutions. Substitute u for x², factor the quadratic in u, then factor or solve each piece. A piece like x² = -4 has no real solutions, so it drops out.

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

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Pick more than one for a sheet that mixes them.

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Sigma Prep · sigmaprep.io/worksheets Name Factoring Quadratic Form Date Period Factor each completely. Factor fourth-degree trinomials with a leading coefficient of 1 $r^{4} - 11 r^{2} + 30$ $k^{4} - 7 k^{2} + 10$ $m^{4} - 8 m^{2} + 15$ $n^{4} - 9 n^{2} + 18$ $n^{4} - 4 n^{2} + 4$ $x^{4} - 5 x^{2} + 6$ $r^{4} - 8 r^{2} + 12$ $r^{4} - 10 r^{2} + 25$ $n^{4} - 6 n^{2} + 9$ $r^{4} - 12 r^{2} + 36$ Solve, or factor completely with a leading coefficient other than 1 $2 x^{4} + 7 x^{2} - 30$ $4 x^{4} + 8 x^{2} + 3$ $2 x^{4} - x^{2} - 10$ $4 x^{4} - 4 x^{2} - 15$ $2 x^{4} - 7 x^{2} + 6$ Find all real solutions: $x^{4} - 20 x^{2} + 64 = 0$ $2 x^{4} + x^{2} - 1$ Find all real solutions: $x^{4} - 29 x^{2} + 100 = 0$ $4 x^{4} + 4 x^{2} + 1$ $2 x^{4} - 13 x^{2} + 6$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Factoring Quadratic Form

Date Period

Factor each completely.

Factor fourth-degree trinomials with a leading coefficient of 1
$r^{4} - 11 r^{2} + 30$
$k^{4} - 7 k^{2} + 10$
$m^{4} - 8 m^{2} + 15$
$n^{4} - 9 n^{2} + 18$
$n^{4} - 4 n^{2} + 4$
$x^{4} - 5 x^{2} + 6$
$r^{4} - 8 r^{2} + 12$
$r^{4} - 10 r^{2} + 25$
$n^{4} - 6 n^{2} + 9$
$r^{4} - 12 r^{2} + 36$

Solve, or factor completely with a leading coefficient other than 1
$2 x^{4} + 7 x^{2} - 30$
$4 x^{4} + 8 x^{2} + 3$
$2 x^{4} - x^{2} - 10$
$4 x^{4} - 4 x^{2} - 15$
$2 x^{4} - 7 x^{2} + 6$
Find all real solutions: $x^{4} - 20 x^{2} + 64 = 0$
$2 x^{4} + x^{2} - 1$
Find all real solutions: $x^{4} - 29 x^{2} + 100 = 0$
$4 x^{4} + 4 x^{2} + 1$
$2 x^{4} - 13 x^{2} + 6$



Sigma Prep · sigmaprep.io/worksheets Name Factoring Quadratic FormAnswer keyVersion 2 Date Period Factor each completely. Factor fourth-degree trinomials with a leading coefficient of 1 $r^{4} - 11 r^{2} + 30$ $(r^{2} - 5) (r^{2} - 6)$ $k^{4} - 7 k^{2} + 10$ $(k^{2} - 2) (k^{2} - 5)$ $m^{4} - 8 m^{2} + 15$ $(m^{2} - 5) (m^{2} - 3)$ $n^{4} - 9 n^{2} + 18$ $(n^{2} - 3) (n^{2} - 6)$ $n^{4} - 4 n^{2} + 4$ ${(n^{2} - 2)}^{2}$ $x^{4} - 5 x^{2} + 6$ $(x^{2} - 2) (x^{2} - 3)$ $r^{4} - 8 r^{2} + 12$ $(r^{2} - 6) (r^{2} - 2)$ $r^{4} - 10 r^{2} + 25$ ${(r^{2} - 5)}^{2}$ $n^{4} - 6 n^{2} + 9$ ${(n^{2} - 3)}^{2}$ $r^{4} - 12 r^{2} + 36$ ${(r^{2} - 6)}^{2}$ Solve, or factor completely with a leading coefficient other than 1 $2 x^{4} + 7 x^{2} - 30$ $(2 x^{2} - 5) (x^{2} + 6)$ $4 x^{4} + 8 x^{2} + 3$ $(2 x^{2} + 1) (2 x^{2} + 3)$ $2 x^{4} - x^{2} - 10$ $(2 x^{2} - 5) (x^{2} + 2)$ $4 x^{4} - 4 x^{2} - 15$ $(2 x^{2} + 3) (2 x^{2} - 5)$ $2 x^{4} - 7 x^{2} + 6$ $(x^{2} - 2) (2 x^{2} - 3)$ Find all real solutions: $x^{4} - 20 x^{2} + 64 = 0$ ${- 4, - 2, 2, 4}$ $2 x^{4} + x^{2} - 1$ $(x^{2} + 1) (2 x^{2} - 1)$ Find all real solutions: $x^{4} - 29 x^{2} + 100 = 0$ ${- 5, - 2, 2, 5}$ $4 x^{4} + 4 x^{2} + 1$ $(2 x^{2} + 1) (2 x^{2} + 1)$ $2 x^{4} - 13 x^{2} + 6$ $(2 x^{2} - 1) (x^{2} - 6)$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Factoring Quadratic FormAnswer keyVersion 2

Date Period

Factor each completely.

Factor fourth-degree trinomials with a leading coefficient of 1
$r^{4} - 11 r^{2} + 30$ $(r^{2} - 5) (r^{2} - 6)$
$k^{4} - 7 k^{2} + 10$ $(k^{2} - 2) (k^{2} - 5)$
$m^{4} - 8 m^{2} + 15$ $(m^{2} - 5) (m^{2} - 3)$
$n^{4} - 9 n^{2} + 18$ $(n^{2} - 3) (n^{2} - 6)$
$n^{4} - 4 n^{2} + 4$ ${(n^{2} - 2)}^{2}$
$x^{4} - 5 x^{2} + 6$ $(x^{2} - 2) (x^{2} - 3)$
$r^{4} - 8 r^{2} + 12$ $(r^{2} - 6) (r^{2} - 2)$
$r^{4} - 10 r^{2} + 25$ ${(r^{2} - 5)}^{2}$
$n^{4} - 6 n^{2} + 9$ ${(n^{2} - 3)}^{2}$
$r^{4} - 12 r^{2} + 36$ ${(r^{2} - 6)}^{2}$

Solve, or factor completely with a leading coefficient other than 1
$2 x^{4} + 7 x^{2} - 30$ $(2 x^{2} - 5) (x^{2} + 6)$
$4 x^{4} + 8 x^{2} + 3$ $(2 x^{2} + 1) (2 x^{2} + 3)$
$2 x^{4} - x^{2} - 10$ $(2 x^{2} - 5) (x^{2} + 2)$
$4 x^{4} - 4 x^{2} - 15$ $(2 x^{2} + 3) (2 x^{2} - 5)$
$2 x^{4} - 7 x^{2} + 6$ $(x^{2} - 2) (2 x^{2} - 3)$
Find all real solutions: $x^{4} - 20 x^{2} + 64 = 0$ ${- 4, - 2, 2, 4}$
$2 x^{4} + x^{2} - 1$ $(x^{2} + 1) (2 x^{2} - 1)$
Find all real solutions: $x^{4} - 29 x^{2} + 100 = 0$ ${- 5, - 2, 2, 5}$
$4 x^{4} + 4 x^{2} + 1$ $(2 x^{2} + 1) (2 x^{2} + 1)$
$2 x^{4} - 13 x^{2} + 6$ $(2 x^{2} - 1) (x^{2} - 6)$

How to do these

Find all real solutions: $x^{4} - 5 x^{2} - 36 = 0$

Treat x² as u: u² - 5u - 36 = (u - 9)(u + 4).
So (x² - 9)(x² + 4) = 0.
x² = 9 gives ±3; x² = -4 has no real solutions.

The answer is ${- 3, 3}$ .

Where students go wrong

Stopping at u. Once u² - 5u - 36 = 0 gives u = 9 and u = -4, each still has to be turned back into x: x² = 9 gives ±3, and x² = -4 gives no real x.

Also called factoring quartics or factoring by substitution.

What you can put on this worksheet

Factor a quadratic in x²: $v^{4} - 5 v^{2} + 6$ → $(v^{2} - 2) (v^{2} - 3)$
Leading coefficients, a GCF, and solving: $2 x^{4} + 5 x^{2} + 2$ → $(2 x^{2} + 1) (x^{2} + 2)$
Cubes and squares inside, grouping, a letter factor, x^n: $18 a^{5} - 21 a^{3} - 72 a$ → $3 a ⁣ (2 a^{2} + 3) (3 a^{2} - 8)$

Questions about these worksheets

Yes. Every factoring quadratic form 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 factoring quadratic form 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 two 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: factor a quadratic in x², leading coefficients, a gcf, and solving and cubes and squares inside, grouping, a letter factor, x^n. Tick as many as you want and set how many of each, or let it spread them evenly.

Stopping at (x² - 4)(x² - 9). Both of those are differences of squares, so the answer is not complete until all four parentheses are there. A minus inside a parenthesis is always worth a second look.

Factoring quadratic form worksheet for Precalculus

Select your difficulty

Factor fourth-degree trinomials with a leading coefficient of 1

Solve, or factor completely with a leading coefficient other than 1

Factor fourth-degree trinomials with a leading coefficient of 1

Solve, or factor completely with a leading coefficient other than 1

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these factoring quadratic form worksheets free?+

Do the factoring quadratic form worksheets come with an answer key?+

Are these factoring quadratic form worksheets printable?+

What grade are these factoring quadratic form worksheets for?+

Can I get a different version of the same factoring quadratic form worksheet?+

Is this a free alternative to Kuta Software for factoring quadratic form?+

Can I choose how hard the factoring quadratic form questions are?+

What kinds of question are on a factoring quadratic form worksheet?+

What do students get wrong with factoring quadratic form?+