The zero product property

About this lesson

Algebra I › Quadratic equations

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The zero product property

Warm-up

Three quick questions from earlier lessons.

Discover

a⋅b=0a \cdot b = 0. What must be true?

  1. a=0a = 0 or b=0b = 0
  2. Both are 0

Key idea

The zero product property.

If A⋅B=0A \cdot B = 0, then A=0A = 0 or B=0B = 0. So to solve a quadratic equation, write it as (factor)(factor) = 0 and set each factor equal to zero. This only works with 0 on one side.

Worked example

Solve (3x−6)(x+4)=0(3x - 6)(x + 4) = 0.

3x−6=03x - 6 = 0, x=2x = 2
First factor.
x+4=0x + 4 = 0, x=−4x = -4
Second factor.

Your turn, guided

Your turn: the non-zero solution of x(x−9)=0x(x - 9) = 0.

99
x − 9 = 0.

Practice ★

  1. Solve x(x−3)=0x(x - 3) = 0. Enter all solutions.
  2. Solve x(x−2)=0x(x - 2) = 0. Enter all solutions.
  1. Solve (4x−6)(x+8)=0(4x - 6)(x + 8) = 0. Enter all solutions.

Watch out!

Practice ★★

  1. To solve (x−3)(x+7)=4(x - 3)(x + 7) = 4, Ana writes “x−3=4x - 3 = 4 or x+7=4x + 7 = 4”. Is that right?
    1. No: the zero product property only works when the product is 0
    2. Yes: set each factor equal to 4
    3. Only the first factor can equal 4
  2. To solve (x−7)(x+6)=17(x - 7)(x + 6) = 17, Ana writes “x−7=17x - 7 = 17 or x+6=17x + 6 = 17”. Is that right?
    1. No: the zero product property only works when the product is 0
    2. Yes: set each factor equal to 17
    3. Only the first factor can equal 17

Exit ticket

Three questions to check you have got it.

  1. Solve (4x−6)(x+8)=0(4x - 6)(x + 8) = 0. Enter all solutions.
  2. Solve (2x−5)(x+9)=0(2x - 5)(x + 9) = 0. Enter all solutions.
  3. Solve (5x−3)(x+8)=0(5x - 3)(x + 8) = 0. Enter all solutions.

Summary card

Zero product property

Product = 0: set each factor to 0.

Watch out: never divide by xx; you lose x=0x = 0.

End of the lesson

Next lesson: Solve quadratic equations by factoring Back to the chapter

Answers: The zero product property

A

  1. 0, 30,\ 3
  2. 0, 20,\ 2

B

  1. 32, −8\frac{3}{2},\ -8

C

  1. No: the zero product property only works when the product is 0
  2. No: the zero product property only works when the product is 0

D

  1. 32, −8\frac{3}{2},\ -8
  2. 52, −9\frac{5}{2},\ -9
  3. 35, −8\frac{3}{5},\ -8

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