Quadratics Word Problems Worksheet: Twelve Stories, Four Shapes
This quadratics word problems worksheet includes balls, fences, ticket prices and triangle for a diverse set of questions. We've put here twelve problems and a full answer key for everyone to practice with.
Published · Last updated · Every answer checked for accuracy before publishing · Free to print for classroom & home use
- Level
- Algebra I and Algebra II, grades 8–11
- Standards
- CCSS.MATH.CONTENT.HSA.CED.A.1, HSA.REI.B.4.b and HSF.IF.B.4
- Includes
- 12 word problems in four sets, a five-item translation warm-up, three color-coded figures, and a full answer key
- Assumes
- Factoring, the quadratic formula, and reading a vertex — the word problems are the new part
- Time
- About 45 minutes, or one set at a time
- Format
- Read on this page or print the PDF — no sign-up
The hardest part of these problems is usually setting up the equation from the short story in the word problem. I have students separate that from the algebra: first identify the unknown and write the equation, then solve it.
- Sketch the situation. It does not have to be good.
- Label one unknown x and write every other quantity in terms of x.
- Write the equation the sentence is describing which could be a height, an area, a revenue, a product.
- Solve it the usual way: factor, use the quadratic formula, or read off the vertex.
- Don't forget to put hte correct unit for your numerical answer. Inches for length or pounds for weight
Want the printable quadratics word problems worksheet?
Seven pages: a translation warm-up, the twelve problems with real space to show work, and the full answer key. No sign-up, no email required.
Download the PDF worksheetPDF · 7 pages · prints on letter paper
Four Setups in This Quadratics Word Problems Worksheet
Most of the quadratic word problems I see in Algebra I and II fall into four common setups.
The −16 is gravity in feet. v₀ is how hard it left, h₀ is where it started.
Ground = 0. Highest point = vertex.
Name one side x, write the other side in terms of x, multiply.
Biggest possible area? Vertex again.
Increasing the price usually decreases the number sold. Multiplying price by quantity produces the quadratic revenue model.
Break-even = roots. Best price = vertex.
Consecutive integers, "a number and its square," legs of a right triangle.
A lot of mistakes on these problems happen after the algebra is finished. A negative time, negative length, or a dimension that does not fit the information in the problem should be rejected. The quadratic can give you two answers but you still have to make sense of it. If something logically is not possible then that quadratics answer can not be the question's answer. Most common ones are negative lengths and weights. Check both answers against the original situation and reject any impossible value.
One Ball, Every Question They Can Ask
A ball leaves a 20-foot roof going up at 32 ft/s: h = −16t² + 32t + 20. That single curve answers everything below.
| Color | Feature | The question that asks for it |
|---|---|---|
| Red | (0, 20) | "From how high was it thrown?" |
| Gold | (1, 36) | "What is the maximum height?" |
| Blue | t = 1 | "When does it peak?" |
| Green | (2.5, 0) | "When does it hit the ground?" |
| Grey | t = −0.5 | Also a root. Also before you threw it. |
Finding that gold point is a whole skill on its own — the vertex and axis of symmetry worksheet drills it, and graphing quadratic functions covers sketching the curve the story describes.
Fence and Barns Questions Are Hidden Rectangles
Forty feet of fence, three sides, barn on the fourth. Each possible value of x gives a different rectangular pen. The maximum area occurs at the vertex of the parabola.
| Color | Feature | What it means on the farm |
|---|---|---|
| Orange | The sketch | Name one side x, then the rest follows |
| Green | x = 0, x = 20 | Pens with no width. Can't use |
| Gold | (10, 200) | Biggest pen: 10 ft by 20 ft |
| Blue | x = 10 | Halfway between the two roots |
If multiplying out x(40 − 2x) and reading the vertex feels shaky, that conversion lives in vertex form and intercept form, with completing the square if you don't get a whole number. If you expand the expression above you'll get -2x^2 +40X. When you use (-b/2a) you can find the h value of 10. Plug that back into the equation and you end up with 200 for k.
Cue Words To Watch Out For
Most of these problems are four phrases you have already seen. Watch out for these words as they will show up in future questions.
| Color | Move | Tool you already have |
|---|---|---|
| Gold | Vertex | x = −b/2a, then plug back in |
| Green | Roots | Factor, or the quadratic formula |
| Red | Constant | Read the constant directly from the equation. |
| Purple | Equal, then 0 | Move everything to one side first |
Reads the problem once
Writes down every number it saw
x = 4 and x = −7
Boxes both. Loses a mark to a fence with negative length.
Sketch it, label one thing x
Write the equation, then solve it
x = 4 (x = −7 rejected: length)
The algebra is the same, but the sketch and final context check make it much easier to reject an impossible solution.
After ten years of watching students work through this unit I've seen multiple different mistakes. It is one of three things: nobody wants to draw a picture, the unknown was never named, or two things in the problem got called x at the same time. The fourth one is forgetting to put your units and check if your number logically makes sense. I hope these questions help address these issues.
Quadratics Word Problems Worksheet: Twelve Practice Problems
| Set | Problem 1 | Problem 2 | Problem 3 |
|---|---|---|---|
| A · THINGS IN THE AIR |
h = −16t² + 32t + 20Thrown off a 20-ft roof. Highest point? When does it land? | h = −16t² + 96tA firework off the ground. At what times is it exactly 128 ft up? | h = −16t² + 24t + 40Off a 40-ft cliff. Highest point, and when does it hit the water? |
| B · THINGS WITH SIDES |
40 ft of fence, three sides, barn on the fourth. What dimensions give the biggest pen, and how big is it? | A rectangle is 3 ft longer than it is wide, area 40 ft². Find both sides. | A 12 in × 16 in photo gets a border of even width all around. Framed area is 320 in². How wide is the border? |
| C · THINGS WITH PRICES |
At $p a ticket the show sells (120 − 4p) tickets. Which price brings in the most money? | P = −x² + 60x − 500How many items to break even? Where is profit biggest? | R = −2x² + 40xWhich prices bring in exactly $150? |
| D · THINGS WITH NUMBERS |
Two consecutive positive integers multiply to 156. Find them. | A right triangle has one leg 7 cm longer than the other and a hypotenuse of 13 cm. Find the legs. | A number plus its square is 72. Find every number that works. |
Set A leans on the formula and the discriminant — those live on the quadratic formula worksheet and the discriminant and number of roots page. Set D is pure factoring practice: see solving quadratics by factoring. Set C's Problem 3 is a two-answer trap of the kind collected on the quadratics error analysis worksheet.
Quadratics Word Problems Worksheet Answer Key
Set A · Things in the air
- A1
- Vertex at t = 1: 36 ft. Lands at t = 2.5 s (the other root, t = −0.5, is rejected)
- A2
- −16t² + 96t = 128 → t² − 6t + 8 = 0 → t = 2 s going up, t = 4 s coming down
- A3
- Vertex at t = 0.75: 49 ft. 2t² − 3t − 5 = 0 → (2t − 5)(t + 1) = 0 → hits the water at t = 2.5 s
Set B · Things with sides
- B1
- A = x(40 − 2x), vertex at x = 10 → 10 ft by 20 ft, area 200 ft²
- B2
- w(w + 3) = 40 → w² + 3w − 40 = 0 → (w + 8)(w − 5) = 0 → 5 ft by 8 ft
- B3
- (12 + 2x)(16 + 2x) = 320 → x² + 14x − 32 = 0 → (x + 16)(x − 2) = 0 → border is 2 in wide
Set C · Things with prices
- C1
- R = p(120 − 4p) = −4p² + 120p, vertex at p = 15 → $15 a ticket, 60 tickets, $900
- C2
- x² − 60x + 500 = 0 → (x − 10)(x − 50) = 0 → break even at 10 and 50 items; biggest profit $400 at x = 30
- C3
- −2x² + 40x = 150 → x² − 20x + 75 = 0 → (x − 5)(x − 15) = 0 → $5 or $15
Set D · Things with numbers
- D1
- n(n + 1) = 156 → n² + n − 156 = 0 → (n + 13)(n − 12) = 0 → 12 and 13
- D2
- x² + (x + 7)² = 169 → x² + 7x − 60 = 0 → (x + 12)(x − 5) = 0 → legs 5 cm and 12 cm
- D3
- x² + x − 72 = 0 → (x + 9)(x − 8) = 0 → x = 8 or x = −9. Both are fine here — nothing in the story forbids a negative
Every answer above was checked for accuracy before publishing, most recently on . If you spot an error, email burketutoringinfremont@outlook.com or text (510) 453-0350 and it will be corrected.
Common Questions
What is a quadratics word problem?
It is a situation described in sentences that turns into a quadratic equation once you name an unknown — a thrown object's height, the area of a rectangle whose sides depend on each other, revenue when price and quantity pull against each other, or a product of two related numbers. Once the equation is written, the solving methods are the same ones students have already practiced.
Why is my kid fine on the equations and lost on the word problems?
A student can be comfortable solving quadratic equations and still struggle to create one from a written situation. I would practice translating the wording separately before combining it with the solving step.
Where does the −16 come from?
It is half of gravity in feet per second squared, rounded. In metric problems you will see −4.9t² instead. Nobody expects a student to derive it; they expect them to recognize it and know the graph opens downward.
When do I throw out one of the two answers?
Whenever the story makes it impossible: negative time, negative length, half a person, a price below zero. Write one line saying why you rejected it. Graders like that line.
What grade level is this quadratics word problems worksheet?
Algebra I covers Sets A, B and D comfortably. Set C fits Algebra I too, though revenue problems tend to show up again in Algebra II and in the SAT's harder math section.
Can they use a graphing calculator?
A graphing calculator is useful for checking the equation once it is written, but the student still has to translate the situation into an equation first.”
Can I use this in my classroom?
Print it, copy it, project it. No sign-up and no attribution required, though a link back is always appreciated.
More Quadratics Practice
If a student is having trouble with the algebra itself, review factoring, the quadratic formula, completing the square, or the discriminant before returning to the word problems: solving by factoring, the quadratic formula, completing the square, or the discriminant. On the picture side: graphing quadratic functions, vertex and axis of symmetry, writing the equation from a graph, and quadratic transformations. Satellite dishes and headlights lead to focus and directrix word problems. Not sure the model is even quadratic? Compare the function families first, then keep everything warm with the quadratics spiral review. To check an answer, graph it in Desmos or GeoGebra, or push the exact roots through WolframAlpha.
Standards and References
These problems were written against the following standards and checked against the following references.
- Setting up an equation from a described situation is CCSS.MATH.CONTENT.HSA.CED.A.1. Standard text: Common Core, High School Algebra: Creating Equations.
- Solving and interpreting the roots is HSA.REI.B.4.b; reading maximums and intercepts in context is HSF.IF.B.4.
- Matched modeling tasks: Illustrative Mathematics, HSA.CED.A.1.
- Problem-solving stance behind the "sketch it first" rule: NRICH, University of Cambridge.
- Alignment and task-quality criteria: Achieve the Core.
- Plain-English refreshers for students: Purplemath on quadratic word problems, Math is Fun on real-world quadratics, Symbolab's step-by-step solver, and Mathway for a second opinion on a single step.
- Answer verification: every solution was substituted back into its original equation, checked against the story for rejected roots, and cross-checked in Desmos and WolframAlpha.
About the Author
Berke has taught K–12 math and science for more than 10 years, logging over 6,000 tutoring hours with 200+ students. He holds a BS in Biochemistry from Washington University in St. Louis and an MS in Bioinformatics from UMGC.
How these are made: problems are prepared from previous class notes and assigned homework, then checked for accuracy before publishing. Corrections come in by email or text and get fixed the same week.
More: tutor profile · LinkedIn · burketutoringinfremont@outlook.com
Page Updates
- — quadratics word problems worksheet published with twelve problems in four sets, three color-coded figures, the printable PDF and the full answer key.
- — added quadratics hub page link
- — Fixed description error link
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