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 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.
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 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.
Example C
A rectangle has sides of length and . What is if the area of the rectangle is 48?
Solution
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.
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:
- 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 and . What is if the area of the rectangle is now 20?
Solution
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.
Practice
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?