Looking at this expression, we can see it is of the form , with , , and . Therefore, we can write it in the form :

A polynominal with the form  multiplies out to , so this leaves two equations: 

Factors of 12 are 2, 6, and 3, 4. In this case, because the product is negative, one root is positive and one is negative. Because the sum is negative, the positive root must have a lower absolute value than the negative root. This leaves us with two possibilities:  2, -6 and 3, -4. Plugging into the sum equation shows the roots to be 3, -4. A check into the original polynominal shows 

When asked to solve for , we are really searching for the roots/-intercepts of the equation. 

In this particular case, our function is already factored for us leaving us with only a few steps to complete the problem. 

Our first step is to set each term equal to , leaving us with...

 and   

The next step is to use our knowledge of order of operations to simply solve for  for each of the above equations...

   Subtract  from both sides

      Divide by 

   Answer #1, our first root/-intercept

  Add  to both sides

      Answer # 2, our second root/-intercept

This problem requires simplification, order of operations, and knowledge of square roots. 

Our goal is to isolate/solve for 

  Divide by  on both sides

     Square root both sides

  **Remember: DO NOT FORGET THAT WHEN WE TAKE A SQUARE ROOT, WE GET A PLUS/MINUS ANSWER.

or

  and  

.

By substituting  - and, subsequently,  this can be rewritten as a quadratic equation, and solved as such:

We are looking to factor the quadratic expression as , replacing the two question marks with integers with product 36 and sum ; these integers are .

Substitute back:

These factors can themselves be factored as the difference of squares:

Set each factor to zero and solve:

The solution set is .

1) Quadratics must be set equal to zero in order to be solved. To do so in this equation, the "8" has to wind up on the left side and combine with any other lone integers. So, multiply out the terms in order to make it possible for the "8" to be added to the other number.

Then combine like terms.

2) Now factor.

1 + 16 = 17

4 + 4 = 8

2 + 8 = 10

3) Pull out common factors, "x" and "8," respectively.

4) Pull out "(x+2)" from both terms.

x = –8, –2

This is a factoring problem, so we need to get all of the variables on one side and set the equation equal to zero. To do this we subtract 128 from both sides to get .

We then notice that all four numbers are divisible by four, so we can simplify the expression to .

Think of the equation in this format to help with the following explanation.

We must then factor to find the solutions for x. To do this we must make a factor tree of c (which is 32 in this case) to find the possible solutions. The possible numbers are 1 * 32, 2 * 16, and 4 * 8.

Since c is negative, we know that our factoring will produce a positive and negative number.

We then look at b to see if the greater number will be positive or negative. Since b is positive, we know that the greater number from our factoring tree will be positive. 

We then use addition and subtraction with the factoring tree to find the numbers that add together to equal b. Remember that the greater number is positive and the lesser number is negative in this example.

Positive 8 and negative 4 equal b. We then plug our numbers into the factored form of .

We know that anything multiplied by 0 is equal to 0, so we plug in the numbers for x which make each equation equal to 0. In this case .

1) Combine like terms and simplify.

No further simplification is possible. The first term has a coefficient that can't be factored away. FOIL requires that all terms be multiplied by each other at some point, so the presence of the coefficient has to be reflected in every step of the factoring.

2) Practically speaking, that means we add an extra step. Multiply the coefficient of the first term by the last term before factoring.

3 * 6 = 18

Factors of 18 include:

1 + 18 = 19

2 + 9 = 11

3) Now pull out the common factor in each of the pairs, "3x" from the first two and "2" from the second two.

4) Pull out the "(x+3)" from both terms.

5) Set both parts equal to zero and solve.

3x + 2 = 0, x = –2/3

x + 3 = 0, x = –3

First, you need to factor the expression. This takes a little bit of trial and error, but even though the leading coefficient is not 1, there's only one way to get 2 (2 x 1) so that is helpful to remember!

Factored, the expression is .

Once you have those expressions, you can set them equal to 0 to get the roots, or solutions to the equation.

When you set .

And when you set

.

Those are your two roots!