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.

This property is unique to 0. Given an equation like $a \boldcdot b = 6$, the factors could be 2 and 3, 1 and 6, -12 and $\text{-}\frac{1}{2}$, $\pi$ and $\frac{6}{\pi}$, or any other of the infinite number of combinations. This type of equation does not give insight into the value of $a$ or $b$.