Zero Product Property

Sometimes we can solve an equation by putting it into Standard Form and then using the Zero Product Property:

The "Standard Form" of an equation is:

(some expression) = 0

In other words, "= 0" is on the right, and everything else is on the left.

Example: Put x 2 = 7 into Standard Form

Standard Form and the Zero Product Property

So let's try it out:

Example: Solve 5(x+3) = 5x(x+3)

It is tempting to divide by (x+3), but that is dividing by zero when x = −3

So instead we can use "Standard Form":

Which can be simplified to:

Then the "Zero Product Property" says:

And the solutions are:

x = 1, or x = −3

And another example:

Example: Solve x 3 = 25x

It is tempting to divide by x, but that is dividing by zero when x = 0

So let's use Standard Form and the Zero Product Property.

Bring all to the left hand side:

x 2 − 25 is a difference of squares, and can be factored into (x − 5)(x + 5) :

Now we can see three possible ways it could end up as zero:

x = 0, or x = 5, or x = −5