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# Solving Problems by Factoring

## Factor and use the zero product rule

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Solving Problems by Factoring

What if you had a triangle in which one leg was one unit shorter than the other and the hypotenuse was 5 units? How could you find the length of the two legs? After completing this Concept, you'll be able to solve real-world applications like this one that involve factoring polynomial equations.

### Guidance

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

#### Example A

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.

Solution

Let $x =$ the length of the short leg of the triangle; then the other leg will measure $x + 3$ .

Use the Pythagorean Theorem: $a^2+b^2=c^2$ , where $a$ and $b$ are the lengths of the legs and $c$ is the length of the hypotenuse. When we substitute the values from the diagram, we get $x^2+(x+3)^2=15^2$ .

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.”

$x^2+x^2+6x+9& =225\\2x^2+6x+9& =225\\2x^2+6x-216 & =0$

Factor out the common monomial: $2(x^2+3x-108)=0$

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:

$-108 &= -12 \cdot 9 && \text{and} && -12 + 9 = -3\\-108 &= 12 \cdot (-9) && \text{and} && 12 + (-9) = 3 \qquad (Correct \ choice)$

We factor the expression as $2(x-9)(x+12)=0$ .

Set each term equal to zero and solve:

$& x-9=0 &&&& x+12=0\\& && \text{or}\\& \underline{\underline{x=9}} &&&& \underline{\underline{x=-12}}$

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

Check: $9^2+12^2=81+144=225=15^2$ , so the answer checks.

#### Example B

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

Solution

Let $x =$ one of the numbers; then $x + 4$ is the other number.

The product of these two numbers is 60, so we can write the equation $x(x+4)=60$ .

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

$x^2+4x &= 60\\x^2+4x-60 &= 0$

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

$-60 &= -4 \cdot 15 && \text{and} && -4 + 15 = 11\\-60 &= 4 \cdot (-15) && \text{and} && 4 + (-15) = -11\\-60 &= -5 \cdot 12 && \text{and} && -5 + 12 = 7\\-60 &= 5 \cdot (-12) && \text{and} && 5 + (-12) = -7\\-60 &= -6 \cdot 10 && \text{and} && -6 + 10 = 4 \qquad (Correct \ choice)\\-60 & = 6 \cdot (-10) && \text{and} && 6 + (-10) = -4$

The expression factors as $(x+10)(x-6)=0$ .

Set each term equal to zero and solve:

$& x+10=0 &&&& x-6=0\\& && \text{or}\\& \underline{\underline{x=-10}} &&&& \underline{\underline{x=6}}$

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

Check: $6 \cdot 10 = 60$ , so the answer checks.

#### Example C

A rectangle has sides of length $x + 5$ and $x - 3$ . What is $x$ if the area of the rectangle is 48?

Solution

Make a sketch of this situation:

Using the formula Area = length $\times$ width, we have $(x+5)(x-3)=48$ .

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

$x^2+2x-15& =48\\x^2+2x-63& =0$

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

$-63 &= -7 \cdot 9 && \text{and} && -7 + 9 = 2 \qquad (Correct \ choice)\\-63 &= 7 \cdot (-9) && \text{and} && 7 + (-9) = -2$

The expression factors as $(x+9)(x-7)=0$ .

Set each term equal to zero and solve:

$& x+9=0 &&&& x-7=0\\& && \text{or}\\& \underline{\underline{x=-9}} &&&& \underline{\underline{x=7}}$

Since we are looking for positive numbers the answer must be $x = 7$ . So the width is $x - 3 = 4$ and the length is $x + 5 = 12$ .

Check: $4 \cdot 12 = 48$ , so the answer checks.

Watch this video for help with the Examples above.

### Vocabulary

• Polynomials can be written in expanded form or in factored form . Expanded form means that you have sums and differences of different terms:
• The factored form of a polynomial means it is written as a product of its factors.
• Zero Product Property: The only way a product is zero is if one or more of the terms are equal to zero: $a\cdot b=0 \Rightarrow a=0 \text{ or } b=0.$
• We say that a polynomial is factored completely when we factor as much as we can and we are unable to factor any more.

### Guided Practice

Consider the rectangle in Example C with sides of length $x + 5$ and $x - 3$ . What is $x$ if the area of the rectangle is now 20?

Solution

Make a sketch of this situation:

Using the formula Area = length $\times$ width, we have $(x+5)(x-3)=20$ .

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

$x^2+2x-15& =20\\x^2+2x-35& =0$

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

$-35 &= -7 \cdot 5 && \text{and} && -7 + 5 = -2 \\-35 &= 7 \cdot (-5) && \text{and} && 7 + (-5) = 2$

The expression factors as $(x+7)(x-5)=0$ .

Set each term equal to zero and solve:

$& x+7=0 &&&& x-5=0\\& && \text{or}\\& \underline{\underline{x=-7}} &&&& \underline{\underline{x=5}}$

Since we are looking for positive numbers the answer must be $x = 5$ . So the width is $x - 3 = 2$ and the length is $x + 5 = 10$ .

Check: $2 \cdot 10 = 20$ , so the answer checks.

### Practice

Solve the following application problems:

1. 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.
2. 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.
3. A rectangle has sides of $x + 2$ and $x - 1$ . What value of $x$ gives an area of 108?
4. A rectangle has sides of $x - 1$ and $x + 1$ . What value of $x$ gives an area of 120?
5. The product of two positive numbers is 120. Find the two numbers if one numbers is 7 more than the other.
6. A rectangle has a 50-foot diagonal. What are the dimensions of the rectangle if it is 34 feet longer than it is wide?
7. Two positive numbers have a sum of 8, and their product is equal to the larger number plus 10. What are the numbers?
8. Two positive numbers have a sum of 8, and their product is equal to the smaller number plus 10. What are the numbers?
9. 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?