Systems of quadratic equations worksheet for Precalculus

Precalculus systems pair a parabola with a line or with a second parabola. Setting the two expressions for y equal gives a quadratic, and the number of real solutions it has tells you whether the graphs meet in two points, one, or none.

Also taught in Algebra 2 and Algebra 1, each at its own level and with its own examples.

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Sigma Prep · sigmaprep.io/worksheets Name Systems of Quadratic Equations Date Period Solve each system. Solve a parabola and line system $\begin{aligned} y & = - 2 x^{2} + 7 x + 36 \\ y & = 5 x - 4 \end{aligned}$ $\begin{aligned} y & = - 3 x^{2} - 19 x + 95 \\ y & = - 7 x - 1 \end{aligned}$ $\begin{aligned} y & = - 3 x^{2} - 26 x \\ y & = x \end{aligned}$ $\begin{aligned} y & = x^{2} \\ y & = 7 x + 8 \end{aligned}$ $\begin{aligned} y & = - x^{2} + 12 x \\ y & = 3 x - 10 \end{aligned}$ $\begin{aligned} y & = - 2 x^{2} + 7 x + 130 \\ y & = x - 10 \end{aligned}$ $\begin{aligned} y & = 3 x^{2} + 21 x - 173 \\ y & = 3 x - 8 \end{aligned}$ $\begin{aligned} y & = 2 x^{2} - 9 x + 5 \\ y & = x + 5 \end{aligned}$ $\begin{aligned} y & = 2 x^{2} - 12 x - 40 \\ y & = 2 x - 4 \end{aligned}$ $\begin{aligned} y & = - 2 x^{2} + 21 x - 75 \\ y & = - 3 x - 11 \end{aligned}$ Solve a system with a parabola $\begin{aligned} y & = x^{2} - x + 4 \\ y & = - x + 13 \end{aligned}$ $\begin{aligned} y & = x^{2} + 2 \\ y & = - 4 x - 3 \end{aligned}$ $\begin{aligned} y & = 2 x^{2} - 4 x + 4 \\ y & = - x^{2} - 13 x + 4 \end{aligned}$ $\begin{aligned} y & = 2 x^{2} - 2 \\ y & = - x^{2} - 3 x + 34 \end{aligned}$ $\begin{aligned} y & = x^{2} + x + 2 \\ y & = 4 x + 6 \end{aligned}$ $\begin{aligned} y & = x^{2} + 3 \\ y & = 2 x + 1 \end{aligned}$ $\begin{aligned} y & = x^{2} + 4 \\ y & = 2 x \end{aligned}$ $\begin{aligned} y & = - x^{2} + 3 x + 5 \\ y & = x^{2} - 3 x - 3 \end{aligned}$ $\begin{aligned} y & = - x^{2} + 3 x - 6 \\ y & = - 2 x^{2} + 7 x - 9 \end{aligned}$ $\begin{aligned} y & = 2 x^{2} - 4 x + 5 \\ y & = - 8 x + 21 \end{aligned}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Systems of Quadratic Equations

Date Period

Solve each system.

Solve a parabola and line system
$\begin{aligned} y & = - 2 x^{2} + 7 x + 36 \\ y & = 5 x - 4 \end{aligned}$
$\begin{aligned} y & = - 3 x^{2} - 19 x + 95 \\ y & = - 7 x - 1 \end{aligned}$
$\begin{aligned} y & = - 3 x^{2} - 26 x \\ y & = x \end{aligned}$
$\begin{aligned} y & = x^{2} \\ y & = 7 x + 8 \end{aligned}$
$\begin{aligned} y & = - x^{2} + 12 x \\ y & = 3 x - 10 \end{aligned}$
$\begin{aligned} y & = - 2 x^{2} + 7 x + 130 \\ y & = x - 10 \end{aligned}$
$\begin{aligned} y & = 3 x^{2} + 21 x - 173 \\ y & = 3 x - 8 \end{aligned}$
$\begin{aligned} y & = 2 x^{2} - 9 x + 5 \\ y & = x + 5 \end{aligned}$
$\begin{aligned} y & = 2 x^{2} - 12 x - 40 \\ y & = 2 x - 4 \end{aligned}$
$\begin{aligned} y & = - 2 x^{2} + 21 x - 75 \\ y & = - 3 x - 11 \end{aligned}$

Solve a system with a parabola
$\begin{aligned} y & = x^{2} - x + 4 \\ y & = - x + 13 \end{aligned}$
$\begin{aligned} y & = x^{2} + 2 \\ y & = - 4 x - 3 \end{aligned}$
$\begin{aligned} y & = 2 x^{2} - 4 x + 4 \\ y & = - x^{2} - 13 x + 4 \end{aligned}$
$\begin{aligned} y & = 2 x^{2} - 2 \\ y & = - x^{2} - 3 x + 34 \end{aligned}$
$\begin{aligned} y & = x^{2} + x + 2 \\ y & = 4 x + 6 \end{aligned}$
$\begin{aligned} y & = x^{2} + 3 \\ y & = 2 x + 1 \end{aligned}$
$\begin{aligned} y & = x^{2} + 4 \\ y & = 2 x \end{aligned}$
$\begin{aligned} y & = - x^{2} + 3 x + 5 \\ y & = x^{2} - 3 x - 3 \end{aligned}$
$\begin{aligned} y & = - x^{2} + 3 x - 6 \\ y & = - 2 x^{2} + 7 x - 9 \end{aligned}$
$\begin{aligned} y & = 2 x^{2} - 4 x + 5 \\ y & = - 8 x + 21 \end{aligned}$



Sigma Prep · sigmaprep.io/worksheets Name Systems of Quadratic EquationsAnswer keyVersion 2 Date Period Solve each system. Solve a parabola and line system $\begin{aligned} y & = - 2 x^{2} + 7 x + 36 \\ y & = 5 x - 4 \end{aligned}$ $(- 4, - 24)$ and $(5, 21)$ $\begin{aligned} y & = - 3 x^{2} - 19 x + 95 \\ y & = - 7 x - 1 \end{aligned}$ $(- 8, 55)$ and $(4, - 29)$ $\begin{aligned} y & = - 3 x^{2} - 26 x \\ y & = x \end{aligned}$ $(- 9, - 9)$ and $(0, 0)$ $\begin{aligned} y & = x^{2} \\ y & = 7 x + 8 \end{aligned}$ $(- 1, 1)$ and $(8, 64)$ $\begin{aligned} y & = - x^{2} + 12 x \\ y & = 3 x - 10 \end{aligned}$ $(- 1, - 13)$ and $(10, 20)$ $\begin{aligned} y & = - 2 x^{2} + 7 x + 130 \\ y & = x - 10 \end{aligned}$ $(- 7, - 17)$ and $(10, 0)$ $\begin{aligned} y & = 3 x^{2} + 21 x - 173 \\ y & = 3 x - 8 \end{aligned}$ $(- 11, - 41)$ and $(5, 7)$ $\begin{aligned} y & = 2 x^{2} - 9 x + 5 \\ y & = x + 5 \end{aligned}$ $(0, 5)$ and $(5, 10)$ $\begin{aligned} y & = 2 x^{2} - 12 x - 40 \\ y & = 2 x - 4 \end{aligned}$ $(- 2, - 8)$ and $(9, 14)$ $\begin{aligned} y & = - 2 x^{2} + 21 x - 75 \\ y & = - 3 x - 11 \end{aligned}$ $(4, - 23)$ and $(8, - 35)$ Solve a system with a parabola $\begin{aligned} y & = x^{2} - x + 4 \\ y & = - x + 13 \end{aligned}$ $(- 3, 16)$ and $(3, 10)$ $\begin{aligned} y & = x^{2} + 2 \\ y & = - 4 x - 3 \end{aligned}$ No real solution $\begin{aligned} y & = 2 x^{2} - 4 x + 4 \\ y & = - x^{2} - 13 x + 4 \end{aligned}$ $(- 3, 34)$ and $(0, 4)$ $\begin{aligned} y & = 2 x^{2} - 2 \\ y & = - x^{2} - 3 x + 34 \end{aligned}$ $(- 4, 30)$ and $(3, 16)$ $\begin{aligned} y & = x^{2} + x + 2 \\ y & = 4 x + 6 \end{aligned}$ $(- 1, 2)$ and $(4, 22)$ $\begin{aligned} y & = x^{2} + 3 \\ y & = 2 x + 1 \end{aligned}$ No real solution $\begin{aligned} y & = x^{2} + 4 \\ y & = 2 x \end{aligned}$ No real solution $\begin{aligned} y & = - x^{2} + 3 x + 5 \\ y & = x^{2} - 3 x - 3 \end{aligned}$ $(- 1, 1)$ and $(4, 1)$ $\begin{aligned} y & = - x^{2} + 3 x - 6 \\ y & = - 2 x^{2} + 7 x - 9 \end{aligned}$ $(1, - 4)$ and $(3, - 6)$ $\begin{aligned} y & = 2 x^{2} - 4 x + 5 \\ y & = - 8 x + 21 \end{aligned}$ $(- 4, 53)$ and $(2, 5)$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Systems of Quadratic EquationsAnswer keyVersion 2

Date Period

Solve each system.

Solve a parabola and line system
$\begin{aligned} y & = - 2 x^{2} + 7 x + 36 \\ y & = 5 x - 4 \end{aligned}$ $(- 4, - 24)$ and $(5, 21)$
$\begin{aligned} y & = - 3 x^{2} - 19 x + 95 \\ y & = - 7 x - 1 \end{aligned}$ $(- 8, 55)$ and $(4, - 29)$
$\begin{aligned} y & = - 3 x^{2} - 26 x \\ y & = x \end{aligned}$ $(- 9, - 9)$ and $(0, 0)$
$\begin{aligned} y & = x^{2} \\ y & = 7 x + 8 \end{aligned}$ $(- 1, 1)$ and $(8, 64)$
$\begin{aligned} y & = - x^{2} + 12 x \\ y & = 3 x - 10 \end{aligned}$ $(- 1, - 13)$ and $(10, 20)$
$\begin{aligned} y & = - 2 x^{2} + 7 x + 130 \\ y & = x - 10 \end{aligned}$ $(- 7, - 17)$ and $(10, 0)$
$\begin{aligned} y & = 3 x^{2} + 21 x - 173 \\ y & = 3 x - 8 \end{aligned}$ $(- 11, - 41)$ and $(5, 7)$
$\begin{aligned} y & = 2 x^{2} - 9 x + 5 \\ y & = x + 5 \end{aligned}$ $(0, 5)$ and $(5, 10)$
$\begin{aligned} y & = 2 x^{2} - 12 x - 40 \\ y & = 2 x - 4 \end{aligned}$ $(- 2, - 8)$ and $(9, 14)$
$\begin{aligned} y & = - 2 x^{2} + 21 x - 75 \\ y & = - 3 x - 11 \end{aligned}$ $(4, - 23)$ and $(8, - 35)$

Solve a system with a parabola
$\begin{aligned} y & = x^{2} - x + 4 \\ y & = - x + 13 \end{aligned}$ $(- 3, 16)$ and $(3, 10)$
$\begin{aligned} y & = x^{2} + 2 \\ y & = - 4 x - 3 \end{aligned}$ No real solution
$\begin{aligned} y & = 2 x^{2} - 4 x + 4 \\ y & = - x^{2} - 13 x + 4 \end{aligned}$ $(- 3, 34)$ and $(0, 4)$
$\begin{aligned} y & = 2 x^{2} - 2 \\ y & = - x^{2} - 3 x + 34 \end{aligned}$ $(- 4, 30)$ and $(3, 16)$
$\begin{aligned} y & = x^{2} + x + 2 \\ y & = 4 x + 6 \end{aligned}$ $(- 1, 2)$ and $(4, 22)$
$\begin{aligned} y & = x^{2} + 3 \\ y & = 2 x + 1 \end{aligned}$ No real solution
$\begin{aligned} y & = x^{2} + 4 \\ y & = 2 x \end{aligned}$ No real solution
$\begin{aligned} y & = - x^{2} + 3 x + 5 \\ y & = x^{2} - 3 x - 3 \end{aligned}$ $(- 1, 1)$ and $(4, 1)$
$\begin{aligned} y & = - x^{2} + 3 x - 6 \\ y & = - 2 x^{2} + 7 x - 9 \end{aligned}$ $(1, - 4)$ and $(3, - 6)$
$\begin{aligned} y & = 2 x^{2} - 4 x + 5 \\ y & = - 8 x + 21 \end{aligned}$ $(- 4, 53)$ and $(2, 5)$

How to do these

Solve: $y = x^{2} - 2 x - 3$ and $y = - x^{2} + 4 x + 5$

Set them equal: x² - 2x - 3 = -x² + 4x + 5.
Collect on one side: 2x² - 6x - 8 = 0, or x² - 3x - 4 = 0, which factors as (x - 4)(x + 1) = 0.
Put each x into either equation: x = -1 gives y = 0 and x = 4 gives y = 5.

The answer is $(- 1, 0), (4, 5)$ .

Where students go wrong

Stopping at the x-values. Each x has to go back into one of the equations to find its y, because the solutions of a system are points, not numbers.

Also called parabola and line intersection, nonlinear systems or quadratic linear systems.

What you can put on this worksheet

A parabola and a line: $\begin{aligned} y & = x^{2} - 3 \\ y & = x - 3 \end{aligned}$ → $(0, - 3)$ and $(1, - 2)$
A circle and a line: $x^{2} + y^{2} = 25$ and $y = x - 5$ → $(5, 0)$ and $(0, - 5)$
Two parabolas, or no solution: $\begin{aligned} y & = 2 x^{2} - 4 x \\ y & = x^{2} - 3 x \end{aligned}$ → $(0, 0)$ and $(1, - 2)$
General-form conics, sideways parabolas, tangents, three points: Solve:
$- 2 y^{2} - x + 29 y - 92 = 0$
$x - y = - 2$ → $(3, 5), (7, 9)$

Questions about these worksheets

Yes. Every systems of quadratic 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 systems of quadratic 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 4: a parabola and a line, a circle and a line, two parabolas, or no solution and general-form conics, sideways parabolas, tangents, three points. Tick as many as you want and set how many of each, or let it spread them evenly.

Stopping at the x values. A system of two curves is answered with points, so each x needs its y. The other slip is substituting back into the parabola, which works but is more arithmetic than using the line.

Systems of quadratic equations worksheet for Precalculus

Select your difficulty

Solve a parabola and line system

Solve a system with a parabola

Solve a parabola and line system

Solve a system with a parabola

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

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