Solving Equations by Factoring
Learn how to solve quadratic equations by factoring them into products of binomials and applying the zero product property.
Definition
Try it now
Worked Examples
Solve:
Identify the equation form
Already in standard form: → , ,
Factor the trinomial
Find two numbers that multiply to 12 and add to 7: and →
Apply zero product property
If , then or → Two equations to solve
Solve each equation
and → or
Answer: or
Common Mistakes
Dividing both sides by (losing the solution )
Why it's wrong: When you divide by , you assume . But might be a valid solution!
Correct: Factor out instead: , giving or
Forgetting to set the equation equal to zero first
Why it's wrong: The zero product property only works when the product equals zero, not any other number.
Correct: Always rewrite as before factoring. For example: becomes
Writing only one solution when there are two
Why it's wrong: Quadratic equations can have two distinct solutions, and both are valid.
Correct: Always check both factors. gives TWO solutions: AND
Sign errors when factoring
Why it's wrong: The signs in the factored form determine the signs of the solutions.
Correct: gives (opposite sign). gives (same sign)
Interactive Visual
Linear Function Explorer
| x | y |
|---|---|
| -2 | -2 |
| -1 | -1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
Balance Scale
Solution: x = 4
Interactive Sandbox
Interactive Grapher
Try these examples:
y = 2x + 1
m=2, b=1
Expression Calculator
Try these:
History
No calculations yet
Practice Problems
15 problemsWhat is the zero product property?
Why It Matters
- Physics: Projectile motion equations (when does the ball hit the ground?)
- Business: Break-even analysis (when does profit equal zero?)
- Engineering: Finding dimensions that satisfy area requirements
- Architecture: Calculating measurements for structural designs
Real World Applications
Projectile Motion
When an object is thrown upward, its height follows a quadratic equation. Finding when it hits the ground means solving for when height equals zero.
Example:
A ball's height is meters after seconds. When does it hit the ground? Solve , or equivalently . Factoring: , so seconds (we ignore since time cannot be negative).
A rocket's height is given by meters, where is time in seconds.
When does the rocket return to the ground?
Step 1: Write the mathematical expression
Set height equal to zero and solve:
Area Problems
Finding dimensions when you know the area often leads to quadratic equations.
Example:
A rectangular garden has length 3 meters more than its width. If the area is 40 square meters, find the dimensions. Let width = . Then: , so . Factoring: , giving meters (width). Length = 8 meters.
A photo frame has width cm and length cm. The area of the frame is 45 square cm.
What are the dimensions of the frame?
Step 1: Write the mathematical expression
Set up and solve the area equation:
Key Takeaways
- 1The zero product property states: if , then or
- 2Always write the equation in standard form () before factoring
- 3Factor the expression, then set each factor equal to zero
- 4Quadratic equations can have zero, one, or two solutions
- 5Check your answers by substituting back into the original equation
- 6Never divide by a variable - you might lose solutions!
Frequently Asked Questions
Glossary
- Zero Product Property
- If , then or (or both)
- Standard Form
- A quadratic equation written as
- Root/Solution
- A value of that makes the equation true
- Factoring
- Writing an expression as a product of simpler expressions