<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation

Solving Problems by Factoring

Factor and use the zero product rule

Atoms Practice
Estimated22 minsto complete
%
Progress
Practice Solving Problems by Factoring
 
 
 
MEMORY METER
This indicates how strong in your memory this concept is
Practice
Progress
Estimated22 minsto complete
%
Practice Now
Turn In
Solving Problems by Factoring

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 \begin{align*}x =\end{align*} the length of the short leg of the triangle; then the other leg will measure \begin{align*}x + 3\end{align*}.

License: CC BY-NC 3.0

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

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.”

\begin{align*}x^2+x^2+6x+9& =225\\ 2x^2+6x+9& =225\\ 2x^2+6x-216 & =0\end{align*}

Factor out the common monomial: \begin{align*}2(x^2+3x-108)=0\end{align*}

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:

\begin{align*}-108 &= -12 \cdot 9 && \text{and} && -12 + 9 = -3\\ -108 &= 12 \cdot (-9) && \text{and} && 12 + (-9) = 3 \qquad (Correct \ choice)\end{align*}

We factor the expression as \begin{align*}2(x-9)(x+12)=0\end{align*}.

Set each term equal to zero and solve:

\begin{align*}& x-9=0 &&&& x+12=0\\ & && \text{or}\\ & \underline{\underline{x=9}} &&&& \underline{\underline{x=-12}}\end{align*}

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

Check: \begin{align*}9^2+12^2=81+144=225=15^2\end{align*}, 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 \begin{align*}x =\end{align*} one of the numbers; then \begin{align*}x + 4\end{align*} is the other number.

The product of these two numbers is 60, so we can write the equation \begin{align*}x(x+4)=60\end{align*}.

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

\begin{align*}x^2+4x &= 60\\ x^2+4x-60 &= 0\end{align*}

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

\begin{align*}-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\end{align*}

The expression factors as \begin{align*}(x+10)(x-6)=0\end{align*}.

Set each term equal to zero and solve:

\begin{align*}& x+10=0 &&&& x-6=0\\ & && \text{or}\\ & \underline{\underline{x=-10}} &&&& \underline{\underline{x=6}}\end{align*}

Since we are looking for positive numbers, the answer must be \begin{align*}x = 6\end{align*}. One number is 6, and the other number is 10.

Check: \begin{align*}6 \cdot 10 = 60\end{align*}, so the answer checks.

Real-World Application: Area

A rectangle has sides of length \begin{align*}x + 5\end{align*} and \begin{align*}x - 3\end{align*}. What is \begin{align*}x\end{align*} if the area of the rectangle is 48?

Make a sketch of this situation:

License: CC BY-NC 3.0

Using the formula Area = length \begin{align*}\times\end{align*} width, we have \begin{align*}(x+5)(x-3)=48\end{align*}.

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

\begin{align*}x^2+2x-15& =48\\ x^2+2x-63& =0\end{align*}

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

\begin{align*}-63 &= -7 \cdot 9 && \text{and} && -7 + 9 = 2 \qquad (Correct \ choice)\\ -63 &= 7 \cdot (-9) && \text{and} && 7 + (-9) = -2\end{align*}

The expression factors as \begin{align*}(x+9)(x-7)=0\end{align*}.

Set each term equal to zero and solve:

\begin{align*}& x+9=0 &&&& x-7=0\\ & && \text{or}\\ & \underline{\underline{x=-9}} &&&& \underline{\underline{x=7}}\end{align*}

Since we are looking for positive numbers the answer must be \begin{align*}x = 7\end{align*}. So the width is \begin{align*}x - 3 = 4\end{align*} and the length is \begin{align*}x + 5 = 12\end{align*}.

Check: \begin{align*}4 \cdot 12 = 48\end{align*}, so the answer checks.

 

 

Example

Example 1

Consider the rectangle in Example C with sides of length \begin{align*}x + 5\end{align*} and \begin{align*}x - 3\end{align*}. What is \begin{align*}x\end{align*} if the area of the rectangle is now 20?

Make a sketch of this situation:

Using the formula Area = length \begin{align*}\times\end{align*} width, we have \begin{align*}(x+5)(x-3)=20\end{align*}.

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

\begin{align*}x^2+2x-15& =20\\ x^2+2x-35& =0\end{align*}

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

\begin{align*}-35 &= -7 \cdot 5 && \text{and} && -7 + 5 = -2 \\ -35 &= 7 \cdot (-5) && \text{and} && 7 + (-5) = 2\end{align*}

The expression factors as \begin{align*}(x+7)(x-5)=0\end{align*}.

Set each term equal to zero and solve:

\begin{align*}& x+7=0 &&&& x-5=0\\ & && \text{or}\\ & \underline{\underline{x=-7}} &&&& \underline{\underline{x=5}}\end{align*}

Since we are looking for positive numbers the answer must be \begin{align*}x = 5\end{align*}. So the width is \begin{align*}x - 3 = 2\end{align*} and the length is \begin{align*}x + 5 = 10\end{align*}.

Check: \begin{align*}2 \cdot 10 = 20\end{align*}, so the answer checks.

Review 

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?

Review (Answers)

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

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Please to create your own Highlights / Notes
Show More

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0

Explore More

Sign in to explore more, including practice questions and solutions for Solving Problems by Factoring.
Please wait...
Please wait...