Quadratics Word Problems Worksheet: Twelve Stories, Four Shapes
This quadratics word problems worksheet turns balls, fences, ticket prices and one suspicious triangle into four familiar setups. Translate first, then solve the equation you already know how to solve. Twelve problems and a full answer key are included.
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
Nobody is stuck on the algebra. They are stuck on sentence three. A quadratics word problem asks you to do two separate jobs — build the equation, then solve it — and mashing them together is where the wheels come off. Build first. Solve second. The solving part you have done a hundred times.
- 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 — a height, an area, a revenue, a product.
- Solve it the usual way: factor, use the quadratic formula, or read off the vertex.
- Throw out any answer the story forbids, then put a unit on the one that survives.
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
Textbooks write hundreds of these. There are four.
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.
Raise the price, sell fewer. That tug-of-war is the parabola.
Break-even = roots. Best price = vertex.
Consecutive integers, "a number and its square," legs of a right triangle.
Almost always factors. Enjoy it while it lasts.
Half the marks lost on these are lost after the algebra is right. A negative time, a negative width, a fence 47 feet long when you only own 40 feet of fence — the quadratic hands you two answers and does not care that one of them is nonsense. Circle the one that survives the story.
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.
One Fence, One Barn, Too Many Rectangles
Forty feet of fence, three sides, barn on the fourth. Every rectangle you can build is one x-value on this curve, and only one of them is the good one.
| 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. Legal, useless |
| 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 as the fallback when the numbers get ugly.
English In, Algebra Out
Most of these problems are four phrases you have already seen. Learn the phrases, not the problems.
| 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 it off. Genuinely that easy |
| 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)
Same algebra, thirty extra seconds, full marks.
After ten years of watching students work through this unit, the stall is almost never the algebra. It is one of three things: nobody drew a picture, the unknown was never named, or two things in the problem got called x at the same time. The fourth, rarer one is finishing the algebra and forgetting the question asked for the height, not the time. These twelve problems are ordered to hit those in that order.
Quadratics Word Problems Worksheet: Twelve Practice Problems
Same twelve as the printable, same order. Sketch before you solve — even a bad sketch.
| 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. The algebra is ordinary; the work is building the equation.
Why is my kid fine on the equations and lost on the word problems?
Because they are two different skills and only one of them gets practiced. Solving x² + 3x − 40 = 0 is a procedure. Turning "3 feet longer than it is wide" into w(w + 3) is reading comprehension with algebra attached. Practice the translating separately — that is what the warm-up page in the PDF is for.
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?
For checking, absolutely. For setting up, it is useless — a calculator cannot read the sentence for you, which is the entire difficulty.
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
Word problems assume the machinery already works. If a set keeps stalling, go back to the machinery: 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 sanity-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.
The four sets on this page are the four story shapes that keep showing up in student homework. Once a student can name which shape they are looking at, the algebra usually takes care of itself.
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
Quadratics turn up in Algebra I at every high school around here — Irvington, American, Mission San Jose, Washington and Kennedy in Fremont, Newark Memorial, and James Logan in Union City. Word problems are the fastest thing to fix in person and the slowest to fix alone, because most of the work is watching a student read. Burke Tutoring offers in-home algebra tutoring in Fremont, Newark, and Union City. Call (510) 453-0350.
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