Now that we know how to factor, we can apply our knowledge to solving quadratic equations that we were unable to solve before.  We will use factoring and the Zero Product Property to solve these equations easily.  Let’s begin by stating the Zero Product Property:

Definition:  Let a and b be any variables, or algebraic expressions.  Then the following is true:

ab=0   if and only if   a=0   or   b=0

The Zero Product Property is critical in our pursuit of solving quadratic equations.  Now let’s go to an example to see how it works.

Example:  Solve the equation \(3{n^2} + 96 = 36n\) by factoring.

Solution:  First manipulate the equation so that it is in standard form

\(3{n^2} + 96 = 36n\)

\(3{n^2} - 36n + 96 = 0\)

We immediately see a factoring opportunity.  All of the terms in the polynomial have a coefficient that is divisible by 3.  We divide both sides by 3 to obtain:

\({n^2} - 12n + 32 = 0\)

This is a less complicated equation.  And we see that we can factor it yet again.  We factor the left side of the equation and we have

\(\left( {n - 8} \right)\left( {n - 4} \right) = 0\)

Good.  We are close to a solution.  Now we implement the Zero Product Property.  We restate the Zero Product Property using our particular expressions:

\(\left( {n - 8} \right)\left( {n - 4} \right) = 0\) if and only if \(n - 8 = 0\) or \(n - 4 = 0\)

So we have that either \(n - 8 = 0\), in which case \(n = 8\), or \(n - 4 = 0\), in which case \(n = 4\). You can check both solutions to see that they are indeed true.

The solution is \(n = 4\) or \(8\). Now we are able to solve quadratic equations that are factorable.  And it’s all because of the Zero Product Property!

Let’s try another example:

Example:  Solve the equation \(2{x^2} = 18\) by factoring.

Solution:  Again, we will put the equation in standard form, factor the left side, then use the Zero Product Property to find the solutions.  We have

\(2{x^2} = 18\)

\(2{x^2} - 18 = 0\), put equation in standard form

\({x^2} - 9 = 0\), divide both sides by 2

\(\left( {x + 3} \right)\left( {x - 3} \right) = 0\), factor the left side

\(x =  - 3or3\), Zero product Property

Bellow you can download some free math worksheets and practice.