### Solving Problems by Factoring

Now that we know most of the factoring strategies for quadratic polynomials, we can apply these methods to solving real world problems.

#### Real-World Application: Right Triangles

One leg of a right triangle is 3 feet longer than the other leg. The hypotenuse is 15 feet. Find the dimensions of the triangle.

Let

Use the Pythagorean Theorem:

In order to solve this equation, we need to get the polynomial in standard form. We must first distribute, collect like terms and **rewrite** in the form “polynomial = 0.”

**Factor** out the common monomial:

To factor the trinomial inside the parentheses, we need two numbers that multiply to -108 and add to 3. It would take a long time to go through all the options, so let’s start by trying some of the bigger factors:

We factor the expression as

**Set each term equal to zero** and **solve:**

It makes no sense to have a negative answer for the length of a side of the triangle, so the answer must be **the short leg is 9 feet and the long leg is 12 feet.**

**Check:**

#### Real-World Application: Finding Unknown Numbers

The product of two positive numbers is 60. Find the two numbers if one number is 4 more than the other.

Let

The product of these two numbers is 60, so we can write the equation

In order to solve we must write the polynomial in standard form. Distribute, collect like terms and **rewrite:**

**Factor** by finding two numbers that multiply to -60 and add to 4. List some numbers that multiply to -60:

The expression factors as

**Set each term equal to zero** and **solve:**

Since we are looking for positive numbers, the answer must be **One number is 6, and the other number is 10.**

**Check:**

#### Real-World Application: Area

A rectangle has sides of length

Make a sketch of this situation:

Using the formula Area = length

In order to solve, we must write the polynomial in standard form. Distribute, collect like terms and **rewrite:**

**Factor** by finding two numbers that multiply to -63 and add to 2. List some numbers that multiply to -63:

The expression factors as

**Set each term equal to zero** and **solve:**

Since we are looking for positive numbers the answer must be **the width is** **and the length is**

**Check:**

### Example

#### Example 1

Consider the rectangle in Example C with sides of length

Make a sketch of this situation:

Using the formula Area = length

In order to solve, we must write the polynomial in standard form. Distribute, collect like terms and **rewrite:**

**Factor** by finding two numbers that multiply to -35 and add to 2. List some numbers that multiply to -35:

The expression factors as

**Set each term equal to zero** and **solve:**

Since we are looking for positive numbers the answer must be **the width is** **and the length is**

**Check:**

### Review

Solve the following application problems:

- One leg of a right triangle is 1 feet longer than the other leg. The hypotenuse is 5. Find the dimensions of the right triangle.
- One leg of a right triangle is 7 feet longer than the other leg. The hypotenuse is 13. Find the dimensions of the right triangle.
- A rectangle has sides of
x+2 and \begin{align*}x - 1\end{align*}. What value of \begin{align*}x\end{align*} gives an area of 108? - A rectangle has sides of \begin{align*}x - 1\end{align*} and \begin{align*}x + 1\end{align*}. What value of \begin{align*}x\end{align*} gives an area of 120?
- The product of two positive numbers is 120. Find the two numbers if one numbers is 7 more than the other.
- A rectangle has a 50-foot diagonal. What are the dimensions of the rectangle if it is 34 feet longer than it is wide?
- Two positive numbers have a sum of 8, and their product is equal to the larger number plus 10. What are the numbers?
- Two positive numbers have a sum of 8, and their product is equal to the smaller number plus 10. What are the numbers?
- Framing Warehouse offers a picture framing service. The cost for framing a picture is made up of two parts: glass costs $1 per square foot and the frame costs $2 per foot. If the frame has to be a square, what size picture can you get framed for $20?

### Review (Answers)

To view the Review answers, open this PDF file and look for section 9.14.