Quadratic Word Problems

Learn to translate real-world situations into quadratic equations and solve them to find meaningful answers.

Advanced25 minLesson
Printable worksheet

Definition

A quadratic word problem is a real-world situation that can be modeled by a quadratic equation of the form:
ax2+bx+c=0ax^2 + bx + c = 0
Common types include:
  • Projectile motion: Objects thrown or launched into the air
  • Area problems: Finding dimensions of rectangles, gardens, or frames
  • Number problems: Finding two numbers with specific relationships
  • Optimization: Finding maximum or minimum values
To solve quadratic word problems:
  1. 1.Read the problem carefully and identify what you need to find
  2. 2.Define a variable for the unknown quantity
  3. 3.Write a quadratic equation based on the relationships described
  4. 4.Solve using factoring, completing the square, or the quadratic formula
  5. 5.Check that your answer makes sense in the real-world context

Try it now

A ball is thrown upward. Its height hh in meters after tt seconds is h=−5t2+20th = -5t^2 + 20t. When does the ball hit the ground?

Worked Examples

A ball is thrown upward from ground level with an initial velocity of 48 feet per second. Its height hh (in feet) after tt seconds is given by h=−16t2+48th = -16t^2 + 48t. When will the ball hit the ground?
1

Understand what we need to find

The ball hits the ground when its height h=0h = 0 → Set h=0h = 0
2

Write the equation

0=−16t2+48t0 = -16t^2 + 48t → −16t2+48t=0-16t^2 + 48t = 0
3

Factor out the GCF

−16t(t−3)=0-16t(t - 3) = 0 → Factored form
4

Apply zero product property

−16t=0-16t = 0 or t−3=0t - 3 = 0 → t=0t = 0 or t=3t = 3
5

Interpret the solutions

t=0t = 0 is when the ball is thrown; t=3t = 3 is when it lands → The ball lands at t=3t = 3 seconds

Common Mistakes

Forgetting to check if solutions make sense in context

Why it's wrong: Mathematical solutions like negative lengths or times before launch are invalid in real-world problems.

Correct: Always interpret your answers: widths must be positive, time must be non-negative, and quantities must be reasonable.

Setting up the equation incorrectly

Why it's wrong: Misreading the relationship between quantities (e.g., confusing 'more than' with 'times').

Correct: Read carefully: '4 more than width' means w+4w + 4, while '4 times the width' means 4w4w.

Using only one solution when both are valid

Why it's wrong: Some problems have two meaningful solutions that both answer the question.

Correct: Check both solutions against the original problem. For example, a ball reaches 20 feet on the way up AND on the way down.

Confusing the vertex formula with the quadratic formula

Why it's wrong: The vertex formula x=−b2ax = -\frac{b}{2a} finds the axis of symmetry, not the roots.

Correct: Use the vertex formula for maximum/minimum problems; use the quadratic formula to find where the parabola crosses the x-axis.

Interactive Visual

Function graph

012345670102030405060th(6.32, 0)(3, 55)

Tap or hover over the graph to read off values.

h = -5t² + 30t + 10
  • Zero (x-intercept): (6.32, 0)
  • Vertex: (3, 55)

Interactive Sandbox

Interactive Grapher

012345670102030405060xy

Tap or hover over the graph to read off values.

Try these examples:

y = -16x^2 + 48x

Expression Calculator

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Practice Problems

16 problems
Problem 1 of 16
Easy
A ball is thrown upward. Its height hh in meters after tt seconds is h=−5t2+20th = -5t^2 + 20t. When does the ball hit the ground?

Why It Matters

Quadratic equations appear everywhere in the real world:
  • Sports: The path of a basketball, soccer ball, or golf ball follows a parabola
  • Architecture: Arches in bridges and buildings are parabolic shapes
  • Business: Profit and revenue models often involve quadratic relationships
  • Engineering: Designing reflectors, antennas, and suspension bridges
  • Physics: Calculating trajectories, acceleration, and energy
Mastering quadratic word problems helps you apply mathematical thinking to solve practical challenges in science, engineering, and everyday life.

Real World Applications

Sports and Projectile Motion

Athletes and coaches use quadratic equations to analyze the trajectory of balls, jumps, and throws.

Example:

A basketball player shoots from the three-point line. The ball's height follows h=−0.05x2+1.2x+6h = -0.05x^2 + 1.2x + 6, where xx is the horizontal distance. Does the ball clear the 10-foot rim at x=20x = 20 feet?

1Try It Yourself

A soccer ball is kicked with height h=−4.9t2+14.7th = -4.9t^2 + 14.7t meters after tt seconds.

How long is the ball in the air?

Step 1: Write the mathematical expression

Set height equal to zero and solve:

Architecture and Design

Architects use quadratic equations to design arches, domes, and optimize space.

Example:

A parabolic arch has the equation y=−0.1x2+2xy = -0.1x^2 + 2x where xx and yy are in meters. The arch spans from x=0x = 0 to x=20x = 20, and its maximum height is at the vertex: y=10y = 10 meters.

2Try It Yourself

An architect designs a rectangular courtyard with a fixed perimeter of 40 meters.

What dimensions maximize the area?

Step 1: Write the mathematical expression

If width = ww, then length = 20−w20 - w. Area = w(20−w)w(20 - w)

Business and Economics

Companies use quadratic models to maximize profit and revenue.

Example:

A company finds that its profit (in thousands of dollars) from selling xx hundred items is P=−2x2+24x−54P = -2x^2 + 24x - 54. Maximum profit occurs when x=6x = 6 hundred items.

3Try It Yourself

A theater sells tickets at 20 dollars. For every 2 dollar increase, they sell 10 fewer tickets. Currently they sell 300 tickets.

What price maximizes revenue?

Step 1: Write the mathematical expression

Let nn = number of 2 dollar increases. Revenue = (price)(tickets)

Key Takeaways

  • 1Quadratic word problems model real situations with equations of the form ax2+bx+c=0ax^2 + bx + c = 0
  • 2Common problem types include projectile motion, area, consecutive integers, and optimization
  • 3Always define your variable clearly and write an equation based on the problem's relationships
  • 4Solve using factoring, completing the square, or the quadratic formula
  • 5Check that your solutions make sense in the real-world context (no negative lengths or impossible times)
  • 6For maximum/minimum problems, find the vertex using x=−b2ax = -\frac{b}{2a}

Frequently Asked Questions

Look for key indicators: area of rectangles (length times width), products of related quantities, projectile motion (objects thrown or dropped), or optimization (finding maximum or minimum values). If the problem involves multiplying a variable by itself, it's likely quadratic.
Look for key indicators: area of rectangles (length times width), products of related quantities, projectile motion (objects thrown or dropped), or optimization (finding maximum or minimum values). If the problem involves multiplying a variable by itself, it's likely quadratic.
Both might be valid! For example, a ball reaches a certain height twice: once going up and once coming down. Read the problem carefully to determine if you need one or both solutions.
Try factoring first if the numbers are simple. Use the quadratic formula when: the equation doesn't factor easily, you have decimals or fractions, or you want to be certain you find all solutions.

Glossary

Quadratic equation
An equation of the form ax2+bx+c=0ax^2 + bx + c = 0 where a≠0a \neq 0
Projectile motion
The curved path of an object thrown or launched, modeled by a quadratic equation
Vertex
The highest or lowest point of a parabola, found at x=−b2ax = -\frac{b}{2a}
Consecutive integers
Integers that follow each other in order (e.g., 5, 6, 7)
Optimization
Finding the maximum or minimum value of a quantity

Formula Card

Quadratic Formula

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Solves any quadratic equation $ax^2 + bx + c = 0$

Vertex Formula

x=−b2ax = -\frac{b}{2a}

Finds the x-coordinate of the vertex (maximum or minimum point)

Projectile Height

h=−12gt2+v0t+h0h = -\frac{1}{2}gt^2 + v_0t + h_0

Height at time $t$ with initial velocity $v_0$ and initial height $h_0$

Today's challengeOne problem, three tries

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