Lesson Objectives
• Demonstrate an understanding of how to solve word problems
• Learn how to set up and solve word problems with quadratic equations

## How to Solve Word Problems with Quadratic Equations

We previously learned how to solve a word problem that involves setting up and solving a linear equation. Here, we will use the same six-step process with problems that involve quadratic equations.

### Six-step method for Applications of Linear Equations in One Variable

1. Read the problem carefully and determine what you are asked to find
• Write down the main objective of the problem
2. Assign a variable to represent the unknown
• If more than one unknown exists, we express the other unknowns in terms of this variable
3. Write out an equation which describes the given situation
4. Solve the equation
5. State the answer using a nice clear sentence
6. Check the result by reading back through the problem
• We need to make sure the answer is reasonable. In other words, if asked how many miles were driven to the store, the answer shouldn't be (-3) as we can't drive a negative amount of miles.
Example 1: Solve each word problem.
The area of a rectangular window is 143 square feet. If the length is 2 feet more than the width, what are the dimensions?
Step 1) Read the problem carefully and determine what you are asked to find:
Here, we are asked to find the dimensions of the window.
Step 2) Assign a variable to represent the unknown: Here, we don't know the length or the width. We do know that the length is 2 feet more than the width.
let x=width of the window
then x + 2=length of the window
Step 3) Write out an equation which describes the given situation:
To do this, let's make a little picture, which can help us to visualize the situation: We should know that the area of a rectangle comes from Length x Width
So we can multiply x, our width by (x + 2), our length, and set this equal to 143. $$x(x+2)=143$$ Step 4) Solve the equation: $$x(x+2)=143$$ $$x^2 + 2x=143$$ $$x^2 + 2x - 143=0$$ $$(x - 11)(x + 13)=0$$ $$x=11, -13$$ Since x represents the width of a window, we can throw out -13 since we can't have a width of -13 feet. This tells us that our only valid solution is 11 for the width and 13 (11 + 2) for the length.
Step 5) State the answer using a nice clear sentence:
The width of the window is 11 feet and the length is 13 feet.
Step 6) Check the result by reading back through the problem:
Reading back through the problem, we can see that our window has an area of 143 square feet.
11 feet x 13 feet=143 square feet
Additionally, we are told that the length is 2 feet more than the width. Here the length is 13 feet, which is 2 feet more than the width.

#### Skills Check:

Example #1

Solve each word problem.

Olivia needs to wash a window in a building that is 16 feet from the ground. To avoid a fence, she decides to rest the ladder against the building. For stability, Olivia decides she should place the ladder 12 feet away from the building. How long of a ladder will Olivia need?

A
20 ft
B
22 ft
C
18 ft
D
29 ft
E
32 ft

Example #2

Solve each word problem.

Two cars left an intersection at the same time. One of the cars traveled directly north, while the other car traveled directly west into heavy traffic. An hour later, they ended up being exactly 41 miles apart from each other. If the car that traveled north traveled 31 miles more than the car that traveled west, how far did each car travel?

Note: NB -> North Bound, WB -> West Bound

A
WB: 15 mi, NB: 13mi
B
WB: 9 mi, NB: 40mi
C
WB: 27 mi, NB: 40mi
D
WB: 6 mi, NB: 37mi
E
WB: 8 mi, NB: 39mi

Example #3

Solve each word problem.

The length of a garden is 12 feet more than the width. If the area of the garden is 325 square feet, what are the dimensions of the garden? Note: (Area is length times width)

A
L: 22, W: 10
B
L: 25, W: 13
C
L: 27, W: 15
D
L: 30, W: 18
E
L: 33, W: 21