What Is the Zero Product Property?

The zero product property is a genius tool for solving equations. A good example is quadratic equations without a constant.

Rule

The Zero Product Property

If the product of $a$ and $b$ is zero, you can use this formula.

a \cdot b = 0 \Leftrightarrow a = 0 or b = 0

Example 1

Solve the quadratic equation $2 x^{2} = 4 x$

First you move all the terms with $x$ over to the left side:

2 x^{2} - 4 x = 0 .

2 and $x$ are common factors. Put them outside the brackets:

2 x \cdot (x - 2) = 0 .

When multiple factors multiplied with each other equals zero, at least one of the factors have to be zero. You can’t multiply values that aren’t equal to zero and get zero as the product!

Here are the factors 2, $x$ and $(x - 2)$ . The number 2 can never be zero, so you can ignore it. You can now set the factors containing $x$ equal to zero and solve the equation. The first factor that can be zero is $x$ :

x = 0

since

2 \cdot 0 \cdot (0 - 2) = 2 \cdot 0 \cdot (- 2) = 0 .

One of the solutions is then $x_{1} = 0$ . The other factor you can set to zero is $(x - 2)$ . You can write this as an equation to find what $x$ has to be.

\begin{array}{l} x - 2 & = 0 \\ x & = 2 \end{array}

The other solution is $x_{2} = 2$ . You know this because

2 \cdot 2 \cdot (2 - 2) = 2 \cdot 2 \cdot 0 = 0 .

The solutions to the equation $2 x^{2} - 4 x = 0$ are then $x_{1} = 0$ and $x_{2} = 2$ .

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