Factoring Word Problems

Apply factoring techniques to solve real-world problems involving area, projectile motion, and number relationships.

Advanced30 minLesson
Printable worksheet

Definition

Factoring word problems require translating real-world situations into quadratic equations, then solving by factoring.
The general approach:
  1. 1.Read the problem carefully and identify what you're solving for
  2. 2.Define your variable (let xx = ...)
  3. 3.Translate the words into an equation
  4. 4.Rearrange into standard form: ax2+bx+c=0ax^2 + bx + c = 0
  5. 5.Factor the quadratic expression
  6. 6.Solve using the zero product property
  7. 7.Check your answer makes sense in context
Common types of factoring word problems:
  • Area problems: Length and width relationships
  • Number problems: Consecutive integers, product relationships
  • Projectile motion: Height as a function of time

Try it now

A rectangular field has length (x+5)(x + 5) meters and width xx meters. Which expression represents its area?

Worked Examples

A rectangular garden has a length that is 3 meters more than its width. If the area is 70 square meters, find the dimensions.
1

Define the variable

Let xx = width of the garden (in meters) → Length = x+3x + 3
2

Write the area equation

Area = length ×\times width\n70=(x+3)×x70 = (x + 3) \times x → 70=x2+3x70 = x^2 + 3x
3

Rearrange to standard form

x2+3x−70=0x^2 + 3x - 70 = 0 → Standard form achieved
4

Factor the quadratic

Find factors of −70-70 that add to 33:\n10×(−7)=−7010 \times (-7) = -70 and 10+(−7)=310 + (-7) = 3 → (x+10)(x−7)=0(x + 10)(x - 7) = 0
5

Apply zero product property

x+10=0x + 10 = 0 or x−7=0x - 7 = 0\nx=−10x = -10 or x=7x = 7
6

Check which solution makes sense

Width cannot be negative, so x=7x = 7 → Width = 7 m, Length = 10 m

Common Mistakes

Forgetting to check if the solution makes sense in context

Why it's wrong: Mathematically, x=−10x = -10 might be a valid solution to the equation, but a width cannot be negative in real life.

Correct: Always ask: Does this answer make sense? Reject negative values for lengths, times, or quantities that must be positive.

Setting up the equation incorrectly

Why it's wrong: Misinterpreting phrases like '3 more than' or '5 less than' leads to wrong equations.

Correct: '3 more than xx' means x+3x + 3. '5 less than xx' means x−5x - 5. Read carefully and double-check your translation.

Forgetting to rearrange to standard form before factoring

Why it's wrong: You can only use factoring effectively when the equation equals zero.

Correct: Always move all terms to one side: ax2+bx+c=0ax^2 + bx + c = 0 before attempting to factor.

Missing the second border in area problems

Why it's wrong: A frame border adds to BOTH sides, so total width = original + 2x2x, not xx.

Correct: For borders or margins, remember they're added to both sides. Draw a diagram to visualize.

Interactive Visual

Function graph

-0.500.511.522.533.5010203040th(0, 0)(3, 0)(1.5, 36)

Tap or hover over the graph to read off values.

h = 48t - 16t^2
  • Zero (x-intercept): (0, 0)
  • Zero (x-intercept): (3, 0)
  • Vertex: (1.5, 36)

Equation Solver

Solve for xx:

x(x+3)=70x\left(x + 3\right) = 70
1
x(x+3)=70x\left(x + 3\right) = 70
Move all terms to one side: ax² + bx + c = 0A quadratic equation is solved from standard form

Tips:

  • • Whatever you do to one side, do to the other
  • • Undo operations in reverse order (PEMDAS backwards)
  • • Check your answer by substituting back

Follow along as we solve the equation step by step.

Interactive Sandbox

Expression Calculator

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Practice Problems

15 problems
Problem 1 of 15
Easy
A rectangular field has length (x+5)(x + 5) meters and width xx meters. Which expression represents its area?

Why It Matters

Factoring word problems connect abstract algebra to practical applications:
  • Architecture: Calculating dimensions when you know the area
  • Sports: Determining when a ball reaches a certain height
  • Business: Finding break-even points in profit equations
  • Engineering: Designing spaces with specific area requirements
Mastering these problems shows you can apply mathematical reasoning to solve real challenges.

Real World Applications

Architecture and Design

Architects often know the area they need but must calculate dimensions based on constraints.

Example:

A room must have 200 square feet of floor space. If the length must be 5 feet more than the width, what are the dimensions?

1Try It Yourself

You're designing a rectangular pool with area 150 square meters. The length is twice the width.

What are the pool dimensions?

Step 1: Write the mathematical expression

Set up: width =x= x, length =2x= 2x, area =x×2x=150= x \times 2x = 150

Sports and Physics

The height of thrown or launched objects follows a quadratic path. Factoring helps find key moments.

Example:

A basketball's height (in feet) is h=−16t2+20t+6h = -16t^2 + 20t + 6. When is the ball at 6 feet?

2Try It Yourself

A rocket's height is modeled by h=−5t2+30th = -5t^2 + 30t. When does it return to the ground?

Find the time when h=0h = 0 (returns to ground).

Step 1: Write the mathematical expression

Solve: −5t2+30t=0-5t^2 + 30t = 0

Business and Economics

Profit equations are often quadratic. Finding break-even points uses factoring.

Example:

A company's profit is modeled by P=−x2+50x−400P = -x^2 + 50x - 400, where xx is units sold. Find the break-even points (where P=0P = 0).

3Try It Yourself

Revenue is R=100x−x2R = 100x - x^2 and costs are C=60x+200C = 60x + 200. Find break-even.

Set profit =0= 0 (Revenue −- Cost =0= 0) and solve.

Step 1: Write the mathematical expression

Profit: (100x−x2)−(60x+200)=0(100x - x^2) - (60x + 200) = 0

Key Takeaways

  • 1Read the problem carefully and identify what you're solving for before writing equations
  • 2Define your variable clearly (let xx = ...) and express other quantities in terms of xx
  • 3Translate word phrases into algebraic expressions: '3 more than' means +3+ 3, 'product' means multiply
  • 4Rearrange the equation to standard form (ax2+bx+c=0ax^2 + bx + c = 0) before factoring
  • 5Apply the zero product property after factoring to find solutions
  • 6Always check if your solutions make sense in the real-world context (no negative lengths or times)

Frequently Asked Questions

Not all word problems result in equations that factor with integers. If you can't find integer factors, you may need to use the quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
Not all word problems result in equations that factor with integers. If you can't find integer factors, you may need to use the quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
Consider the context. For physical quantities like length, width, time, or count, negative solutions are usually rejected. Choose the solution that makes sense in the real-world situation.
Area = length ×\times width. When both dimensions depend on the same variable xx (like xx and x+3x + 3), multiplying them creates an x2x^2 term, making it quadratic.

Glossary

Word problem
A mathematical problem presented in everyday language that requires translation into equations
Standard form
A quadratic equation written as ax2+bx+c=0ax^2 + bx + c = 0
Zero product property
If ab=0ab = 0, then a=0a = 0 or b=0b = 0 (used after factoring to find solutions)
Consecutive integers
Integers that follow each other in order, like 5, 6, 7 or represented as xx, x+1x+1, x+2x+2
Projectile motion
The path of an object thrown or launched, typically following a parabolic (quadratic) trajectory
Today's challengeOne problem, three tries

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