Lesson 4
Solving Quadratic Equations with the Zero Product Property
 Let’s find solutions to equations that contain products that equal zero.
4.1: Math Talk: Solve These Equations
What values of the variables make each equation true?
\(6 + 2a = 0\)
\(7b=0\)
\(7(c5)=0\)
\(g \boldcdot h=0\)
4.2: Take the Zero Product Property Out for a Spin
For each equation, find its solution or solutions. Be prepared to explain your reasoning.
 \(x3=0\)
 \(x+11=0\)
 \(2x+11=0\)
 \(x(2x+11)=0\)
 \((x3)(x+11)=0\)
 \((x3)(2x+11)=0\)
 \(x(x+3)(3x4)=0\)
 Use factors of 48 to find as many solutions as you can to the equation \((x3)(x+5)=48\).
 Once you think you have all the solutions, explain why these must be the only solutions.
4.3: Revisiting a Projectile
We have seen quadratic functions modeling the height of a projectile as a function of time.
Here are two ways to define the same function that approximates the height of a projectile in meters, \(t\) seconds after launch:
\(\displaystyle h(t)=\text5t^2+27t+18 \qquad \qquad h(t)=(\text5t3)(t6)\)
 Which way of defining the function allows us to use the zero product property to find out when the height of the object is 0 meters?
 Without graphing, determine at what time the height of the object is 0 meters. Show your reasoning.
Summary
The zero product property says that if the product of two numbers is 0, then one of the numbers must be 0. In other words, if \(a\boldcdot b=0,\) then either \(a=0\) or \(b=0\). This property is handy when an equation we want to solve states that the product of two factors is 0.
Suppose we want to solve \(m(m+9)=0\). This equation says that the product of \(m\) and \((m+9)\) is 0. For this to be true, either \(m=0\) or \(m+9=0\), so both 0 and 9 are solutions.
Here is another equation: \((u2.345)(14u+2)=0\). The equation says the product of \((u2.345)\) and \((14u+2)\) is 0, so we can use the zero product property to help us find the values of \(u\). For the equation to be true, one of the factors must be 0.
 For \(u2.345=0\) to be true, \(u\) would have to be 2.345.
 For \(14u+2=0\) or \(14u = \text2\) to be true, \(u\) would have to be \(\text\frac{2}{14}\) or \(\text\frac17\).
The solutions are 2.345 and \(\text\frac17\).
In general, when a quadratic expression in factored form is on one side of an equation and 0 is on the other side, we can use the zero product property to find its solutions.
Glossary Entries

zero product property
The zero product property says that if the product of two numbers is 0, then one of the numbers must be 0.