Zero Product Property

Learn how the zero product property helps you solve quadratic equations by setting each factor equal to zero.

Intermediate25 minLesson
Printable worksheet

Definition

The Zero Product Property states that if the product of two or more factors equals zero, then at least one of the factors must be zero.
If a⋅b=0, then a=0 or b=0 (or both)\text{If } a \cdot b = 0, \text{ then } a = 0 \text{ or } b = 0 \text{ (or both)}
This property is the key to solving factored quadratic equations:
If (x−3)(x+2)=0\text{If } (x - 3)(x + 2) = 0
Then x−3=0 or x+2=0\text{Then } x - 3 = 0 \text{ or } x + 2 = 0
So x=3 or x=−2\text{So } x = 3 \text{ or } x = -2
Why does this work? The only way to multiply numbers and get zero is if at least one of them is zero. Try it: 5×0=05 \times 0 = 0, 0×(−7)=00 \times (-7) = 0, but 3×4=12≠03 \times 4 = 12 \neq 0.

Try it now

What does the Zero Product Property state?

Worked Examples

Solve (x−5)(x+3)=0(x - 5)(x + 3) = 0
1

Apply Zero Product Property

If (x−5)(x+3)=0(x - 5)(x + 3) = 0, then one factor must equal zero → x−5=0x - 5 = 0 or x+3=0x + 3 = 0
2

Solve first equation

x−5=0⇒x=5x - 5 = 0 \Rightarrow x = 5
3

Solve second equation

x+3=0⇒x=−3x + 3 = 0 \Rightarrow x = -3
4

State both solutions

The equation has two solutions → x=5x = 5 or x=−3x = -3

Common Mistakes

Dividing both sides by a variable factor

Why it's wrong: If you divide by xx, you lose the solution x=0x = 0. For example, in x(x−3)=0x(x-3) = 0, dividing by xx gives only x=3x = 3, missing x=0x = 0.

Correct: Never divide by a variable. Instead, use the zero product property to get ALL solutions.

Applying to equations not equal to zero

Why it's wrong: The property ONLY works when the product equals zero. (x−2)(x+1)=6(x-2)(x+1) = 6 does NOT mean x−2=6x-2 = 6 or x+1=6x+1 = 6.

Correct: First move all terms to one side to get =0= 0, then factor, then apply the property.

Forgetting to factor first

Why it's wrong: The equation x2−5x+6=0x^2 - 5x + 6 = 0 is not yet in factored form. You cannot apply the property directly.

Correct: Factor to (x−2)(x−3)=0(x-2)(x-3) = 0 first, then apply the zero product property.

Missing solutions when one factor is just xx

Why it's wrong: In x(x−4)=0x(x-4) = 0, students sometimes only find x=4x = 4 and forget that x=0x = 0 is also a solution.

Correct: Remember: xx by itself is a factor too! If x=0x = 0, the product is zero.

Interactive Visual

Function graph

-6-4-202468-20-1001020xy(-3, 0)(5, 0)(1, -16)

Tap or hover over the graph to read off values.

y = (x - 5)(x + 3)
  • Zero (x-intercept): (-3, 0)
  • Zero (x-intercept): (5, 0)
  • Vertex: (1, -16)

Equation Solver

Solve for xx:

(x−5)(x+3)=0\left(x - 5\right)\left(x + 3\right) = 0
1
(x−5)(x+3)=0\left(x - 5\right)\left(x + 3\right) = 0
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

Try these:

History

No calculations yet

Practice Problems

17 problems
Problem 1 of 17
Easy
What does the Zero Product Property state?

Why It Matters

The zero product property is your most powerful tool for solving polynomial equations:
  • Finding x-intercepts: The solutions tell you where a parabola crosses the x-axis
  • Physics problems: Finding when a projectile hits the ground (height = 0)
  • Business applications: Finding break-even points (profit = 0)
  • Engineering: Finding equilibrium points in systems
Once you can factor a polynomial, this property instantly gives you the solutions!

Real World Applications

Projectile Motion

When you throw a ball, its height follows a quadratic path. Finding when height equals zero tells you when it lands.

Example:

A ball's height is modeled by h=−16t2+48th = -16t^2 + 48t feet. Setting h=0h = 0 and factoring: −16t(t−3)=0-16t(t - 3) = 0, so t=0t = 0 (launch) or t=3t = 3 seconds (landing).

1Try It Yourself

A rocket's height is given by h=−5t2+20th = -5t^2 + 20t meters, where tt is time in seconds.

When does the rocket hit the ground?

Step 1: Write the mathematical expression

Factor and solve −5t2+20t=0-5t^2 + 20t = 0:

Business Break-Even Analysis

Companies use quadratic equations to model profit. The zero product property finds break-even points where profit equals zero.

Example:

If profit P=x2−100xP = x^2 - 100x euros where xx is units sold, then x(x−100)=0x(x - 100) = 0 gives break-even at x=0x = 0 or x=100x = 100 units.

2Try It Yourself

A company's profit function is P=x2−50xP = x^2 - 50x euros, where xx is the number of products sold.

At what sales levels does the company break even?

Step 1: Write the mathematical expression

Factor and solve x2−50x=0x^2 - 50x = 0:

Key Takeaways

  • 1The Zero Product Property: if ab=0ab = 0, then a=0a = 0 or b=0b = 0
  • 2To use it, the equation must be in the form (factor)(factor) = 0
  • 3Set each factor equal to zero and solve for the variable
  • 4Never divide by a variable - you might lose solutions
  • 5The number of solutions equals the number of distinct linear factors

Frequently Asked Questions

Zero is special! If ab=12ab = 12, we can't conclude anything about aa or bb individually (could be 3×43 \times 4, 2×62 \times 6, 1×121 \times 12, etc.). But for ab=0ab = 0, at least one factor MUST be zero.
Zero is special! If ab=12ab = 12, we can't conclude anything about aa or bb individually (could be 3×43 \times 4, 2×62 \times 6, 1×121 \times 12, etc.). But for ab=0ab = 0, at least one factor MUST be zero.
Like (x−3)2=0(x-3)^2 = 0? You still set x−3=0x - 3 = 0, giving x=3x = 3. This is called a 'repeated root' or 'double root' - the parabola touches but doesn't cross the x-axis there.
Yes! A polynomial of degree nn can have up to nn solutions. For example, x(x−1)(x+2)(x−3)=0x(x-1)(x+2)(x-3) = 0 has four solutions: x=0,1,−2,3x = 0, 1, -2, 3.

Glossary

Zero Product Property
If ab=0ab = 0, then a=0a = 0 or b=0b = 0 (or both)
Factor
An expression that is multiplied with others to form a product
Root/Solution
A value of xx that makes the equation true
Double root
A solution that appears twice, from a repeated factor like (x−a)2(x-a)^2
Today's challengeOne problem, three tries

More in This Topic

Introduction to Factoring

Intermediate25 min

Factoring Out the GCF

Intermediate25 min

Factoring by Grouping

Intermediate25 min

Factoring Trinomials (a=1)

Intermediate25 min