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.

Solve each equation by factoring.

This free worksheet contains 10 assignments each with 24 questions with answers.**Example of one question:**

**Watch below how to solve this example:**

Solve each equation by factoring.

This free worksheet contains 10 assignments each with 24 questions with answers.**Example of one question:**

**Watch below how to solve this example:**

Solve each equation by factoring.

This free worksheet contains 10 assignments each with 24 questions with answers.**Example of one question:**

**Watch below how to solve this example:**