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Solving Systems of Equations with Quadratics - Printable Mathematics Worksheets - Wayground

Solving Systems of Equations with Quadratics

16 questions

10th - 11th Grade

Mathematics

Solving Systems of Equations with Quadratics is a Grade 10–11 mathematics worksheet focused on finding solutions to systems that pair a linear equation with a quadratic equation. Across 16 items—15 multiple-choice questions and one multiple-select question—students solve systems by substitution, identify the first substitution step, evaluate a coordinate in a system, determine the number of solutions, and recognize solutions from graphs. The worksheet also asks students to identify intersection points, interpret ordered pairs, and select all valid solutions when a graph shows more than one intersection. These varied questions connect algebraic and graphical methods while reinforcing that a system’s solutions are the points shared by both equations. Use this free printable worksheet with an answer key for independent practice, classwork, homework, or a focused assessment of linear-quadratic systems.

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Worksheets

Solving Systems of Equations with Quadratics

Total questions: 16

Name
Class
Date
1.
Find the solution:
y = x2 - 4x + 6
y = x + 2
a)
(3, 1) and (6, 4)
b)
(-1, 1) and (4, 2)
c)
(1, 3) and (4, 6)
d)
(1, 4) and (3, 6)
2.
Determine the solution to the given linear-quadratic function.
a)
(0, 0)
b)
(3, 0)
c)
(2, -3)
d)
(-3, 2)
3.
Determine the number of solutions to the given linear-quadratic system.
a)
0 solutions
b)
1 solution
c)
2 solutions
d)
3 solutions
4.

Which of the following represents the first step in solving by substitution:

y = x2 + 3x - 5

y = x + 3

a)

x2 - 3x + 5 = x - 3

b)

x2 + 3x - 5 = x + 3

c)

x2 + 3x - 5 + x + 3 = 0

d)

x2 + 3x - 5 = 0

5.

Which of the following graphs correctly represents:

y=x2

y=-2x+1

a)

b)

c)

d)

6.

What are the solutions to the graphed system?

a)

(0,2) and (3,1)

b)

(-2,0) and (2,0)

c)

(0,-2) and (3,1)

d)

(2,2) and (1,0)

7.
Given the system of equations:
y = x2 + 3x - 5
y = x + 3
When x = -4, what is the value of y?
a)
-1
b)
1
c)
7
d)
-33
8.
Find the solution.
a)
(0,7) and (0,-3)
b)
(2,6) and (7,4)
c)
(3,0) and (7,4)
d)
(-1,5) and (-3,9)
9.
Solve the system of linear and quadratic equations using substitution:
y = 5
y = 2x2 - 16x + 29
a)
(6, 5) and (2, 5)
b)
(2, 6)
c)
(6, 2)
d)
(5, 6) and (5, 2)
10.
Find the solution(s).
a)
(-1,7) and (0,5)
b)
(-4,2) and (-5,8)
c)
(-2,6) and (10,15)
d)
(-2,4) and (6,12)
11.
Find the solution(s).
a)
(3,18) and (-4, 15)
b)
(5,2) and (5,-9)
c)
(-4,-4)
d)
(-4,2)
12.

Solve the system.

y = x2 - x - 90

y = x + 30

a)

(10, 2) and (-10, 42)

b)

(12, 42) and (-10, 20)

c)

(42, 1) and (20, -1)

d)

(2, 12) and (4, 5)

13.

What are the solutions to the system of equations graphed below? (More then one correct answer).

a)

(2, 0)

b)

(0, 4)

c)

(-2,0)

d)

(0, -4)

e)

No Solution

14.

Identify the solution to the system graphed here.

a)

(1, 5) and (-4, 0)

b)

(1, 5) and (-2, 0)

c)

(1, 5) and (0, -4)

d)

No solution

15.

x2+y2+x−2y−21=0 and x2+y2+5x−2y−9=0x^2+y^2+x-2y-21=0\ and\ x^2+y^2+5x-2y-9=0

Find the points of intersections

a)

(3,3), (3,5)

b)

(−3,−3), (−3,5)\left(-3,-3\right),\ \left(-3,5\right)

c)

(-3,-3), (3,-5)

d)

(-3,3), (-3,-5)

16.

Find the solutions if x2−y−2=0 and 2x2+y−6x−7=0Find\ the\ solutions\ if\ x^2-y-2=0\ and\ 2x^2+y-6x-7=0

a)

(3,7), (-1,-1)

b)

(7,3), (1,-1)

c)

(-3,-7), (-1,-1)

d)

(3,7), (1,1)

Answer Key

Solving Systems of Equations with Quadratics

Total questions: 16

1.
c) (1, 3) and (4, 6)
2.
c) (2, -3)
3.
c) 2 solutions
4.
b)

x2 + 3x - 5 = x + 3

5.
b)
6.
c)

(0,-2) and (3,1)

7.
a) -1
8.
c) (3,0) and (7,4)
9.
a) (6, 5) and (2, 5)
10.
d) (-2,4) and (6,12)
11.
d) (-4,2)
12.
b)

(12, 42) and (-10, 20)

13.
a)

(2, 0)

, c)

(-2,0)

14.
a)

(1, 5) and (-4, 0)

15.
b)

(−3,−3), (−3,5)\left(-3,-3\right),\ \left(-3,5\right)

16.
a)

(3,7), (-1,-1)

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FAQs

What does this worksheet cover?

This worksheet covers solving systems made up of one linear equation and one quadratic equation. Its 16 items have students use substitution, evaluate values in a system, determine whether there are zero, one, or two solutions, and identify intersection points from graphs, including a multiple-select item with more than one correct solution.

How can I use this worksheet to teach solving linear-quadratic systems?

Introduce the worksheet by modeling how setting the two expressions for y equal creates a single equation in x, then show how each resulting x-value produces an ordered-pair solution. As students work, ask them to compare substitution results with the intersection points shown in the graph-based items and explain why a linear and quadratic graph can have zero, one, or two shared points.

What mistakes should I watch for when students complete this worksheet?

Watch for students reversing or misreading ordered pairs, reporting only one intersection when a system has two, or choosing “no solution” without checking the graph or algebra. In the substitution items, students may skip the initial step of setting the linear and quadratic expressions equal, and in the multiple-select question they may select only one valid coordinate instead of all correct solutions.

How do I assign and grade this worksheet?

This worksheet is available as a printable PDF and as a digital quiz on Wayground, and it includes a complete answer key for checking the substitution, graph-reading, and ordered-pair responses. For printable submissions, you can scan student work with the Wayground for Teachers app to help grade responses efficiently.

How can I differentiate this linear-quadratic systems worksheet for my students?

Use the worksheet’s Advanced Settings to adjust font spacing and font size so equation choices, coordinate pairs, and graph-based questions are easier to read. You can also apply a dyslexia-friendly font or translate the worksheet into another language while keeping the substitution and intersection-point tasks intact.

Where can I find more worksheets like this on Wayground?

Wayground offers a comprehensive collection of free printable worksheets and practice problems across all subjects, with downloadable PDFs and answer keys to support essential skills and concepts. Browse more resources at https://wayground.com/en-us/worksheets.

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