Factoring Word Problems
Apply factoring techniques to solve real-world problems involving area, projectile motion, and number relationships.
Definition
- Area problems: Length and width relationships
- Number problems: Consecutive integers, product relationships
- Projectile motion: Height as a function of time
Try it now
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.
Define the variable
Let = width of the garden (in meters) → Length =
Write the area equation
Area = length width\n →
Rearrange to standard form
→ Standard form achieved
Factor the quadratic
Find factors of that add to :\n and →
Apply zero product property
or \n or → or
Check which solution makes sense
Width cannot be negative, so → Width = 7 m, Length = 10 m
Answer: The garden is 7 meters wide and 10 meters long.
Common Mistakes
Forgetting to check if the solution makes sense in context
Why it's wrong: Mathematically, 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 ' means . '5 less than ' means . 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: 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 + , not .
Correct: For borders or margins, remember they're added to both sides. Draw a diagram to visualize.
Interactive Visual
Linear Function Explorer
Click on the bar to change the fraction
Interactive Sandbox
Expression Calculator
Try these:
History
No calculations yet
Practice Problems
15 problemsA rectangular field has length meters and width meters. Which expression represents its area?
Why It Matters
- 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
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?
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 , length , area
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 . When is the ball at 6 feet?
A rocket's height is modeled by . When does it return to the ground?
Find the time when (returns to ground).
Step 1: Write the mathematical expression
Solve:
Business and Economics
Profit equations are often quadratic. Finding break-even points uses factoring.
Example:
A company's profit is modeled by , where is units sold. Find the break-even points (where ).
Revenue is and costs are . Find break-even.
Set profit (Revenue Cost ) and solve.
Step 1: Write the mathematical expression
Profit:
Key Takeaways
- 1Read the problem carefully and identify what you're solving for before writing equations
- 2Define your variable clearly (let = ...) and express other quantities in terms of
- 3Translate word phrases into algebraic expressions: '3 more than' means , 'product' means multiply
- 4Rearrange the equation to standard form () 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
Glossary
- Word problem
- A mathematical problem presented in everyday language that requires translation into equations
- Standard form
- A quadratic equation written as
- Zero product property
- If , then or (used after factoring to find solutions)
- Consecutive integers
- Integers that follow each other in order, like 5, 6, 7 or represented as , ,
- Projectile motion
- The path of an object thrown or launched, typically following a parabolic (quadratic) trajectory