The area of a square is

### Factoring Quadratic Functions

When we add a number in front of the **cannot** use the shortcut. First, let’s try FOIL-ing when the coefficients in front of the

Let's multiply

We can still use FOIL.

FIRST

OUTSIDE

INSIDE

LAST

Combining all the terms together, we get:

Now, let’s work backwards and factor a trinomial to get two factors. Remember, you can always check your work by multiplying the final factors together.

Factor

This is a factorable trinomial. When there is a coefficient, or number in front of

Now, we can see, we need the two factors of -12 that also add up to -1.

Factors |
Sum |
---|---|

11 | |

1, -12 | -11 |

2, -6 | -4 |

-2, 6 | 4 |

-3, 4 | 1 |

The factors that work are 3 and -4. Now, take these factors and rewrite the

Next, group the first two terms together and the last two terms together and pull out any common factors.

What is in the parenthesis is *the same*. We now have two terms that both have

The factors of

Now, let's factor

Let’s make the steps we followed in the previous problem a little more concise.

Step 1: Find

Step 2: Rewrite the trinomial with the

Step 3: Group the first two and second two terms together, find the GCF and factor again.

Alternate Method: What happens if we list

This tells us it does not matter which

Let's factor

Let’s use the steps from the previous problem, but we are going to add an additional step at the beginning.

Step 1: Look for any common factors. Pull out the GCF of all three terms, if there is one.

This will make it much easier for you to factor what is inside the parenthesis.

Step 2: Using what is inside the parenthesis, find

The factors of -60 that add up to -11 are -15 and 4.

Step 3: Rewrite the trinomial with the

Step 4: Group the first two and second two terms together, find the GCF and factor again.

### Examples

#### Example 1

Earlier, you were asked to find the dimensions of the square.

The dimensions of a square are its length and its width, so we need to factor the area \begin{align*}9x^2 + 24x + 16\end{align*}.

We need to multiply together \begin{align*}a\end{align*} and \begin{align*}c\end{align*} (from \begin{align*}ax^2+bx+c\end{align*}) and then find the two numbers whose product is \begin{align*}ac\end{align*} and whose sum is \begin{align*}b\end{align*}.

Now we can see that we need the two factors of 144 that also add up to 24. Testing the possibilities, we find that \begin{align*}12 \cdot 12 = 144\end{align*} and \begin{align*}12 + 12 = 24\end{align*}.

Now, take these factors and rewrite the \begin{align*}x-\end{align*}term expanded using 12 and 12.

\begin{align*}& \quad \ 9x^2 {\color{red}+24x}+16\\ & \qquad \ \ {\color{red} \swarrow}{\color{red} \searrow}\\ & 9x^2{\color{red}+12x+12x}+16\end{align*}

Next, group the first two terms together and the last two terms together and pull out any common factors.

\begin{align*}& (9x^2+12x)+(12x+16)\\ & 3x(3x+4)+4(3x+4)\end{align*}

We now have two terms that both have \begin{align*}(3x+4)\end{align*} as factor. Pull this factor out.

The factors of \begin{align*}9x^2+24x+16\end{align*} are \begin{align*}(3x + 4)(3x + 4)\end{align*}, which are also the dimensions of the square.

#### Example 2

Multiply \begin{align*}(4x-3)(3x + 5)\end{align*}.

FOIL: \begin{align*}(4x-3)(3x + 5) = 12x^2 +20x-9x-15 = 12x^2+11x-15\end{align*}

**Factor the following quadratics, if possible.**

#### Example 3

#### \begin{align*}15x^2-4x-3\end{align*}

Use the steps from the examples above. There is no GCF, so we can find the factors of \begin{align*}ac\end{align*} that add up to \begin{align*}b\end{align*}.

\begin{align*}15 \cdot -3 = -45 \end{align*} The factors of -45 that add up to -4 are -9 and 5.

\begin{align*}& 15x^2 {\color{red}-4x}-3\\ & (15x^2 {\color{red}-9x)}+({\color{red}5x}-3)\\ & 3x(5x-3)+1(5x-3)\\ & (5x-3)(3x+1)\end{align*}

#### Example 4

\begin{align*}3x^2+6x-12\end{align*}

\begin{align*}3x^2+6x-12\end{align*} has a GCF of 3. Pulling this out, we have \begin{align*}3(x^2+2x-6)\end{align*}. There is no number in front of \begin{align*}x^2\end{align*}, so we see if there are any factors of -6 that add up to 2. There are not, so this trinomial is not factorable.

#### Example 5

\begin{align*}24x^2-30x-9\end{align*}

\begin{align*}24x^2-30x-9\end{align*} also has a GCF of 3. Pulling this out, we have \begin{align*}3(8x^2-10x-3)\end{align*}. \begin{align*}ac = -24\end{align*}. The factors of -24 than add up to -10 are -12 and 2.

\begin{align*}& 3(8x^2{\color{red}-10x}-3)\\ & 3 \left[(8x^2{\color{red}-12x})+({\color{red}2x}-3)\right]\\ & 3 \left[ 4x(2x-3)+1(2x-3)\right]\\ & 3(2x-3)(4x+1)\end{align*}

#### Example 6

\begin{align*}4x^2+4x-48\end{align*}

\begin{align*}4x^2+4x-48\end{align*}. has a GCF of 4. Pulling this out, we have \begin{align*}4(x^2+x-12)\end{align*}. This trinomial does not have a number in front of \begin{align*}x^2\end{align*}, so we can use the shortcut we are familiar with. What are the factors of -12 that add up to 1?

\begin{align*}& 4(x^2+x-12)\\ & 4(x+4)(x-3)\end{align*}

### Review

Multiply the following expressions.

- \begin{align*}(2x-1)(x + 5)\end{align*}
- \begin{align*}(3x + 2)(2x-3)\end{align*}
- \begin{align*}(4x + 1)(4x-1)\end{align*}

Factor the following quadratic equations, if possible. If they cannot be factored, write *not factorable*. Don’t forget to look for any GCFs first.

- \begin{align*}5x^2+18x+9\end{align*}
- \begin{align*}6x^2-21x\end{align*}
- \begin{align*}10x^2-x-3\end{align*}
- \begin{align*}3x^2+2x-8\end{align*}
- \begin{align*}4x^2+8x+3\end{align*}
- \begin{align*}12x^2-12x-18\end{align*}
- \begin{align*}16x^2-6x-1\end{align*}
- \begin{align*}5x^2-35x+60\end{align*}
- \begin{align*}2x^2+7x+3\end{align*}
- \begin{align*}3x^2+3x+27\end{align*}
- \begin{align*}8x^2-14x-4\end{align*}
- \begin{align*}10x^2+27x-9\end{align*}
- \begin{align*}4x^2+12x+9\end{align*}
- \begin{align*}15x^2+35x\end{align*}
- \begin{align*}6x^2-19x+15\end{align*}
- Factor \begin{align*}x^2-25\end{align*}. What is \begin{align*}b\end{align*}?
- Factor \begin{align*}9x^2-16\end{align*}. What is \begin{align*}b\end{align*}? What types of numbers are \begin{align*}a\end{align*} and \begin{align*}c\end{align*}?

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 5.2.