Quadratic Word Problems With Worked Examples

Quadratics show up whenever something is squared: the area of a shape, the height of a thrown ball, or revenue when price and sales both change. The algebra is the same every time; the skill is setting up the equation and deciding which solution makes sense.

Quick idea

Translate the story into an equation, rearrange it to equal zero, solve by factoring or the quadratic formula, then reject any answer that is impossible in the real situation.

Quadratic formula solver

Type the coefficients of ax² + bx + c = 0. You get the discriminant, both solutions (exact and decimal), and every step.

abc

Two real solutions: x ≈ 3 and x ≈ 2

x=3\ \text{or}\ x=2

Step 1.Start from standard form and read off a, b, and c (signs included).
$\begin{gathered}x^2-5x+6=0\\ a=1,\ b=-5,\ c=6\end{gathered}$
Step 2.Substitute every value into the quadratic formula, keeping negatives in parentheses.
$x=\frac{-(-5)\pm\sqrt{(-5)^2-4(1)(6)}}{2(1)}$
Step 3.Work out the discriminant, the part under the square root.
$b^2-4ac=25-24=1$
Step 4.The discriminant is positive, so there are two different real solutions.
$x=\frac{5\pm\sqrt{1}}{2}=\frac{5\pm 1}{2}$
Step 5.Split the ± into a plus version and a minus version.
$\begin{aligned}x_1&=\frac{5+1}{2}\approx 3\\ x_2&=\frac{5-1}{2}\approx 2\end{aligned}$
Step 6.Exact form, fully simplified:
$x=3\ \text{or}\ x=2$
Step 7.Bonus: the vertex of the parabola sits halfway between the roots, at x = −b/(2a).
$\left(2.5,\ -0.25\right)$

The four types you will see most

Type	Typical equation	Key idea
Projectile (thrown or dropped objects)	$h=-16t^2+v_0t+h_0$ (feet) or $h=-4.9t^2+v_0t+h_0$ (meters)	Lands when $h=0$ ; highest at the vertex
Area	length × width = area	Write both sides in terms of one variable
Number puzzles	$n(n+1)=\dots$	Consecutive integers differ by 1
Revenue / profit	price × quantity sold	Maximum value is at the vertex

Steps

Define a variable and write what it stands for, with units.
Build the equation from the story, then rearrange it to standard form = 0.
Solve by factoring, completing the square, or the quadratic formula.
Check each solution in the story and keep only the ones that make sense.

Worked example

A ball is thrown from a 48-foot platform. Its height after t seconds is h = −16t² + 32t + 48. When does it hit the ground?

Full solution

-16t^2+32t+48=0\quad\Rightarrow\quad t^2-2t-3=0\quad\Rightarrow\quad (t-3)(t+1)=0\quad\Rightarrow\quad t=3

Step-by-step walkthrough

Step 1

The ball hits the ground when its height is 0, so set the expression equal to 0.

-16t^2+32t+48=0

Step 2

Divide every term by −16 to make the numbers smaller.

t^2-2t-3=0

Step 3

Factor: two numbers that multiply to −3 and add to −2 are −3 and 1.

(t-3)(t+1)=0

Step 4

t = 3 or t = −1. Time cannot be negative, so the ball lands after 3 seconds.

t=3

More worked examples

Example 2: Area of a rectangle

A garden is 3 m longer than it is wide, and its area is 40 m². Find its dimensions.

Step 1. Let w be the width in metres. Then the length is w + 3.

w(w+3)=40

Step 2. Expand and rearrange to standard form.

w^2+3w-40=0

Step 3. Factor: 8 × (−5) = −40 and 8 + (−5) = 3.

(w+8)(w-5)=0

Answer: w = 5 (reject w = −8, since a width cannot be negative). The garden is 5 m by 8 m.

Example 3: A picture frame

A 10 cm by 8 cm photo has a frame of equal width x all around. The total area is 120 cm². How wide is the frame?

Step 1. The frame adds x to each side, so each dimension grows by 2x.

(10+2x)(8+2x)=120

Step 2. Expand and simplify.

4x^2+36x+80=120\ \Rightarrow\ x^2+9x-10=0

Step 3. Factor.

(x+10)(x-1)=0

Answer: x = 1 cm (x = −10 is impossible).

Example 4: Maximum revenue

A shop sells 200 − 10p items when the price is p dollars. What price gives the most revenue?

Step 1. Revenue is price times quantity.

R=p(200-10p)=-10p^2+200p

Step 2. The maximum is at the vertex.

p=-\frac{b}{2a}=-\frac{200}{2(-10)}=10

Step 3. Find the revenue at that price.

R=-10(10)^2+200(10)=1000

Answer: Charge $10 for a maximum revenue of $1,000.

Example 5: When it will not factor

A ball's height is $h=-16t^2+48t+6$ feet. When does it hit the ground?

Step 1. Set h = 0 and divide by −2 to simplify.

8t^2-24t-3=0

Step 2. This does not factor, so use the quadratic formula.

t=\frac{24\pm\sqrt{576+96}}{16}=\frac{24\pm\sqrt{672}}{16}

Step 3. Evaluate. √672 ≈ 25.92.

t\approx\frac{24+25.92}{16}\approx3.12\quad\text{or}\quad t\approx-0.12

Answer: About 3.12 seconds (the negative time is rejected).

Which answer should you keep?

A quadratic usually gives two solutions, but the story may only allow one. Reject:

negative times (before the event started),
negative or zero lengths, widths, and prices,
non-whole numbers when counting people or objects.

Always say why you rejected an answer — teachers often give marks for it.

Maximum and minimum questions use the vertex

"Maximum height", "largest area", and "greatest profit" are asking for the vertex, not the roots. For $y=ax^2+bx+c$ , the best input is $x=-\frac{b}{2a}$ , and the best value is $y$ at that input.

Common mistakes to avoid

Do not report both solutions automatically. Negative times, negative lengths, and fractional numbers of people usually have to be rejected.
Forgetting to rearrange to standard form (= 0) before factoring or using the formula.
Confusing 'when does it hit the ground' (roots) with 'how high does it go' (vertex).
Leaving off units, or answering with the variable's value when the question asked for something else (like the length instead of the width).

Practice problems

A rectangle's length is 3 cm more than its width, and its area is 40 cm². Find its dimensions.
The product of two consecutive positive integers is 72. Find them.
A ball's height is h = −16t² + 64t. What is its maximum height?

Answers

5 cm by 8 cm
8 and 9
64 feet, at t = 2 seconds

Frequently asked questions

How do I know a word problem needs a quadratic?

Look for something being squared or two unknown quantities being multiplied: area of a shape, products of numbers, height of a thrown object, or price times quantity sold.

Why do I get a negative answer?

The equation does not know about the real situation, so it often gives one impossible solution. Reject negative times and lengths, and explain why.

Should I factor or use the quadratic formula?

Try factoring first. If the numbers do not work out quickly, or the answer is clearly a decimal, use the quadratic formula — it always works.

Ask PennPaper to explain it live

If the steps make sense but you still feel stuck, start a PennPaper lesson and ask the tutor to draw the problem on the whiteboard. Seeing the symbols move step by step is often what makes the concept click.

Related guides

Quick idea

Translate the story into an equation, rearrange it to equal zero, solve by factoring or the quadratic formula, then reject any answer that is impossible in the real situation.

Quadratic formula solver

Type the coefficients of ax² + bx + c = 0. You get the discriminant, both solutions (exact and decimal), and every step.

abc

Two real solutions: x ≈ 3 and x ≈ 2

x=3\ \text{or}\ x=2

Step 1.Start from standard form and read off a, b, and c (signs included).
$\begin{gathered}x^2-5x+6=0\\ a=1,\ b=-5,\ c=6\end{gathered}$
Step 2.Substitute every value into the quadratic formula, keeping negatives in parentheses.
$x=\frac{-(-5)\pm\sqrt{(-5)^2-4(1)(6)}}{2(1)}$
Step 3.Work out the discriminant, the part under the square root.
$b^2-4ac=25-24=1$
Step 4.The discriminant is positive, so there are two different real solutions.
$x=\frac{5\pm\sqrt{1}}{2}=\frac{5\pm 1}{2}$
Step 5.Split the ± into a plus version and a minus version.
$\begin{aligned}x_1&=\frac{5+1}{2}\approx 3\\ x_2&=\frac{5-1}{2}\approx 2\end{aligned}$
Step 6.Exact form, fully simplified:
$x=3\ \text{or}\ x=2$
Step 7.Bonus: the vertex of the parabola sits halfway between the roots, at x = −b/(2a).
$\left(2.5,\ -0.25\right)$

The four types you will see most

Type	Typical equation	Key idea
Projectile (thrown or dropped objects)	$h=-16t^2+v_0t+h_0$ (feet) or $h=-4.9t^2+v_0t+h_0$ (meters)	Lands when $h=0$ ; highest at the vertex
Area	length × width = area	Write both sides in terms of one variable
Number puzzles	$n(n+1)=\dots$	Consecutive integers differ by 1
Revenue / profit	price × quantity sold	Maximum value is at the vertex

Steps

Define a variable and write what it stands for, with units.
Build the equation from the story, then rearrange it to standard form = 0.
Solve by factoring, completing the square, or the quadratic formula.
Check each solution in the story and keep only the ones that make sense.

Worked example

A ball is thrown from a 48-foot platform. Its height after t seconds is h = −16t² + 32t + 48. When does it hit the ground?

Full solution

-16t^2+32t+48=0\quad\Rightarrow\quad t^2-2t-3=0\quad\Rightarrow\quad (t-3)(t+1)=0\quad\Rightarrow\quad t=3

Step-by-step walkthrough

Step 1

The ball hits the ground when its height is 0, so set the expression equal to 0.

-16t^2+32t+48=0

Step 2

Divide every term by −16 to make the numbers smaller.

t^2-2t-3=0

Step 3

Factor: two numbers that multiply to −3 and add to −2 are −3 and 1.

(t-3)(t+1)=0

Step 4

t = 3 or t = −1. Time cannot be negative, so the ball lands after 3 seconds.

t=3

More worked examples

Example 2: Area of a rectangle

A garden is 3 m longer than it is wide, and its area is 40 m². Find its dimensions.

Step 1. Let w be the width in metres. Then the length is w + 3.

w(w+3)=40

Step 2. Expand and rearrange to standard form.

w^2+3w-40=0

Step 3. Factor: 8 × (−5) = −40 and 8 + (−5) = 3.

(w+8)(w-5)=0

Answer: w = 5 (reject w = −8, since a width cannot be negative). The garden is 5 m by 8 m.

Example 3: A picture frame

A 10 cm by 8 cm photo has a frame of equal width x all around. The total area is 120 cm². How wide is the frame?

Step 1. The frame adds x to each side, so each dimension grows by 2x.

(10+2x)(8+2x)=120

Step 2. Expand and simplify.

4x^2+36x+80=120\ \Rightarrow\ x^2+9x-10=0

Step 3. Factor.

(x+10)(x-1)=0

Answer: x = 1 cm (x = −10 is impossible).

Example 4: Maximum revenue

A shop sells 200 − 10p items when the price is p dollars. What price gives the most revenue?

Step 1. Revenue is price times quantity.

R=p(200-10p)=-10p^2+200p

Step 2. The maximum is at the vertex.

p=-\frac{b}{2a}=-\frac{200}{2(-10)}=10

Step 3. Find the revenue at that price.

R=-10(10)^2+200(10)=1000

Answer: Charge $10 for a maximum revenue of $1,000.

Example 5: When it will not factor

A ball's height is $h=-16t^2+48t+6$ feet. When does it hit the ground?

Step 1. Set h = 0 and divide by −2 to simplify.

8t^2-24t-3=0

Step 2. This does not factor, so use the quadratic formula.

t=\frac{24\pm\sqrt{576+96}}{16}=\frac{24\pm\sqrt{672}}{16}

Step 3. Evaluate. √672 ≈ 25.92.

t\approx\frac{24+25.92}{16}\approx3.12\quad\text{or}\quad t\approx-0.12

Answer: About 3.12 seconds (the negative time is rejected).

Which answer should you keep?

A quadratic usually gives two solutions, but the story may only allow one. Reject:

negative times (before the event started),
negative or zero lengths, widths, and prices,
non-whole numbers when counting people or objects.

Always say why you rejected an answer — teachers often give marks for it.

Maximum and minimum questions use the vertex

"Maximum height", "largest area", and "greatest profit" are asking for the vertex, not the roots. For $y=ax^2+bx+c$ , the best input is $x=-\frac{b}{2a}$ , and the best value is $y$ at that input.

Common mistakes to avoid

Do not report both solutions automatically. Negative times, negative lengths, and fractional numbers of people usually have to be rejected.
Forgetting to rearrange to standard form (= 0) before factoring or using the formula.
Confusing 'when does it hit the ground' (roots) with 'how high does it go' (vertex).
Leaving off units, or answering with the variable's value when the question asked for something else (like the length instead of the width).

Practice problems

A rectangle's length is 3 cm more than its width, and its area is 40 cm². Find its dimensions.
The product of two consecutive positive integers is 72. Find them.
A ball's height is h = −16t² + 64t. What is its maximum height?

Answers

5 cm by 8 cm
8 and 9
64 feet, at t = 2 seconds

Frequently asked questions

How do I know a word problem needs a quadratic?

Look for something being squared or two unknown quantities being multiplied: area of a shape, products of numbers, height of a thrown object, or price times quantity sold.

Why do I get a negative answer?

The equation does not know about the real situation, so it often gives one impossible solution. Reject negative times and lengths, and explain why.

Should I factor or use the quadratic formula?

Try factoring first. If the numbers do not work out quickly, or the answer is clearly a decimal, use the quadratic formula — it always works.

Quick idea

Quadratic formula solver

The four types you will see most

Steps

Worked example

Full solution

Step-by-step walkthrough

Step 1

Step 2

Step 3

Step 4

More worked examples

Example 2: Area of a rectangle

Example 3: A picture frame

Example 4: Maximum revenue

Example 5: When it will not factor

Which answer should you keep?

Maximum and minimum questions use the vertex

Common mistakes to avoid

Practice problems

Answers

Frequently asked questions

How do I know a word problem needs a quadratic?

Why do I get a negative answer?

Should I factor or use the quadratic formula?

Ask PennPaper to explain it live

Related guides

Related Articles

How to Add Fractions With Unlike Denominators

How to Subtract Fractions Step by Step

How to Multiply Fractions

Ready to try PennPaper?

Quick idea

Quadratic formula solver

The four types you will see most

Steps

Worked example

Full solution

Step-by-step walkthrough

Step 1

Step 2

Step 3

Step 4

More worked examples

Example 2: Area of a rectangle

Example 3: A picture frame

Example 4: Maximum revenue

Example 5: When it will not factor

Which answer should you keep?

Maximum and minimum questions use the vertex

Common mistakes to avoid

Practice problems

Answers

Frequently asked questions

How do I know a word problem needs a quadratic?

Why do I get a negative answer?

Should I factor or use the quadratic formula?

Ask PennPaper to explain it live

Related guides

Related Articles

How to Add Fractions With Unlike Denominators

How to Subtract Fractions Step by Step

How to Multiply Fractions

Ready to try PennPaper?