# 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?

$$6 + 2a = 0$$

$$7b=0$$

$$7(c-5)=0$$

$$g \boldcdot h=0$$

### 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.

1. $$x-3=0$$
2. $$x+11=0$$
3. $$2x+11=0$$
4. $$x(2x+11)=0$$
5. $$(x-3)(x+11)=0$$
6. $$(x-3)(2x+11)=0$$
7. $$x(x+3)(3x-4)=0$$

1. Use factors of 48 to find as many solutions as you can to the equation $$(x-3)(x+5)=48$$.
2. 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:

$$\displaystyle h(t)=\text-5t^2+27t+18 \qquad \qquad h(t)=(\text-5t-3)(t-6)$$

1. 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?
2. 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: $$(u-2.345)(14u+2)=0$$. The equation says the product of $$(u-2.345)$$ 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 $$u-2.345=0$$ to be true, $$u$$ would have to be 2.345.
• For $$14u+2=0$$ or $$14u = \text-2$$ 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.