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Illustrative
Mathematics
A-REI Zero Product Property 3
Alignments to Content Standards: A-REI.A.1
Task
The Zero Product Property (ZPP) states that if the product of two numbers is zero, then
at least one of the numbers is zero. In symbols, if ab = 0, then a = 0 or b = 0. We can
use this property when we solve equations where a product is 0. For each equation
below, use the ZPP to find all solutions. Explain each step in your reasoning.
a. x(13 − 4x)
b. 7(y + 12)
= 0,
= 0,
c. (x − 19)(x + 3)
= 0,
d. (y − 6)(3z − 4)
= 0.
IM Commentary
This task is part of a series of tasks that lead students to understand and apply the zero
product property to solving quadratic equations. The emphasis is on using the
structure of a factorable expression to help find its solutions (rather than memorizing
steps without understanding). Teachers should feel free to skip any tasks in the series
that students have already mastered.
In previous tasks, students stated and proved the ZPP. In this particular task, we are
trying to get students to use the property to reason about solutions to equations given
in factored form. In tasks that follow in this series, students will first need to factor
expressions before applying the ZPP to solve quadratic equations.
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Illustrative
Mathematics
Note: the use of "or" in the ZPP is the mathematician's inclusive "or." That is, when we
say, " a = 0 or b = 0," the "or both a and b are 0" is implied. Alternatively, one could
state this verbally: "If the product of two numbers is zero, then at least one of them has
to be zero." The last equation in the problem is a good place to highlight the meaning
of "or" in the ZPP, since in order for this equation to be true, y could be 6, z could be 4 ,
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but also both could be true at the same time.
Links to other tasks in this series:
Zero Product Property 1
Zero Product Property 2
Zero Product Property 4
Edit this solution
Solution
a. If x(13 − 4x) = 0, then by the Zero Product Property (ZPP), either x = 0 or
13 − 4x = 0. So x = 0 is one solution to this equation. Solving 13 − 4x = 0, we get
x = 13 for the other solution.
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b. If 7(y + 12) = 0, then by the ZPP, either 7 = 0, which is not true, or
Hence the only solution to this equation is y = −12.
c. If (x − 19)(x + 3) = 0, then by the ZPP, either x − 19
solutions to the equation are x = 19 and x = −3.
y + 12 = 0.
= 0 or x + 3 = 0. So the
d. If (y − 6)(3z − 4) = 0, then by the ZPP, y − 6 = 0 or 3z − 4 = 0. So the solutions to
the equation are y = 6 (and z can be any value) or z = 4 (and y can be any value).
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A-REI Zero Product Property 3
Typeset May 4, 2016 at 21:40:19. Licensed by Illustrative Mathematics under a
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .
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