A Product That Makes Zero

If two numbers multiply to zero, then one of them is zero. There is no other way to get a product of zero. $5\cdot 3$ is 15, not zero, and $0.1\cdot 0.1$ is 0.01, not zero. A product is zero only when one of its factors is zero.

Now look at $\left(x-2\right)\left(x+5\right) = 0$. The left side is a product and the right side is zero, so one of the two factors must be zero. Either $x-2 = 0$ or $x+5 = 0$. Solve each of these small equations: $x = 2$ or $x = -5$.

The zero product rule says that a product equals zero when at least one of its factors equals zero, and only then. To solve an equation with a product on one side and zero on the other, set each factor equal to zero. Then solve each of the small equations you get. For $\left(x-2\right)\left(x+5\right) = 0$, the two small equations give $x = 2$ or $x = -5$, and both are solutions of the original equation.