Solving factored equations

In my last post, you saw a technique to solve equations when one side consists of factors (things multiplied together) and the other side is zero. Generally, if you have any number of factors with unknowns in them, the only way that the equation can be solved is by setting each factor to zero and solving for the unknown.

That is, if you have say four unknown numbers which multiplied together equal zero: abcd = 0, the only way that this can be true is if any of the unknown numbers are zero.

Let’s have a more complex example. Consider

(x – 4)(x + 7)(x-3)(x+5) = 0

Here are four expressions multiplied together, each with an unknown number x. This can be solved by setting each factor to zero as you can only get zero through multiplication if any of the things multiplied are zero themselves:

x – 4 = 0  ⇒  x = 4

x + 7 = 0  ⇒  x = -7

x – 3 = 0  ⇒  x = 3

x + 5 = 0  ⇒  x = -5

So there are four solutions to this equation.

This is in fact a technique frequently used to solve equations with powers of x

But there are many ways to factor and those skills will be covered in future posts.