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9.14: Solving Problems by Factoring

Difficulty Level: At Grade Created by: CK-12
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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.

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CK-12 Foundation: 0914S Solving Real-World Problems By Factoring

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: a2+b2=c2, 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 x2+(x+3)2=152.

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.”

x2+x2+6x+92x2+6x+92x2+6x216=225=225=0

Factor out the common monomial: 2(x2+3x108)=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:

108108=129=12(9)andand12+9=312+(9)=3(Correct choice)

We factor the expression as 2(x9)(x+12)=0.

Set each term equal to zero and solve:

x9=0x=9orx+12=0x=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: 92+122=81+144=225=152, 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:

x2+4xx2+4x60=60=0

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

606060606060=415=4(15)=512=5(12)=610=6(10)andandandandandand4+15=114+(15)=115+12=75+(12)=76+10=4(Correct choice)6+(10)=4

The expression factors as (x+10)(x6)=0.

Set each term equal to zero and solve:

x+10=0x=10orx6=0x=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: 610=60, so the answer checks.

Example C

A rectangle has sides of length x+5 and x3. What is x if the area of the rectangle is 48?

Solution

Make a sketch of this situation:

Using the formula Area = length × width, we have (x+5)(x3)=48.

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

x2+2x15x2+2x63=48=0

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

6363=79=7(9)andand7+9=2(Correct choice)7+(9)=2

The expression factors as (x+9)(x7)=0.

Set each term equal to zero and solve:

x+9=0x=9orx7=0x=7

Since we are looking for positive numbers the answer must be x=7. So the width is x3=4 and the length is x+5=12.

Check: 412=48, so the answer checks.

Watch this video for help with the Examples above.

CK-12 Foundation: Solving Real-World Problems by Factoring

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: ab=0a=0 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 x3. What is x if the area of the rectangle is now 20?

Solution

Make a sketch of this situation:

Using the formula Area = length × width, we have (x+5)(x3)=20.

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

x2+2x15x2+2x35=20=0

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

3535=75=7(5)andand7+5=27+(5)=2

The expression factors as (x+7)(x5)=0.

Set each term equal to zero and solve:

x+7=0x=7orx5=0x=5

Since we are looking for positive numbers the answer must be x=5. So the width is x3=2 and the length is x+5=10.

Check: 210=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 \begin{align*}x + 2\end{align*} and \begin{align*}x - 1\end{align*}. What value of \begin{align*}x\end{align*} gives an area of 108?
  4. 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?
  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?

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Oct 01, 2012
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