### 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 the length of the short leg of the triangle; then the other leg will measure .

Use the Pythagorean Theorem: , where and are the lengths of the legs and is the length of the hypotenuse. When we substitute the values from the diagram, we get .

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 . That means **the short leg is 9 feet and the long leg is 12 feet.**

**Check:** , so the answer checks.

#### 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 one of the numbers; then is the other number.

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:** , so the answer checks.

#### Real-World Application: Area

A rectangle has sides of length and . What is if the area of the rectangle is 48?

Make a sketch of this situation:

Using the formula Area = length width, we have .

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 . So **the width is** **and the length is** .

**Check:** , so the answer checks.

### Example

#### Example 1

Consider the rectangle in Example C with sides of length and . What is if the area of the rectangle is now 20?

Make a sketch of this situation:

Using the formula Area = length width, we have .

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 . So **the width is** **and the length is** .

**Check:** , so the answer checks.

### 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 and . What value of gives an area of 108?
- A rectangle has sides of and . What value of 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.