Quadratic Equations: example 7.81

A rectangular garden has an area $15$ square feet. The length of the garden is two feet more than the width. Find the length and width of the garden.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

A rectangular sign has an area of 30 square feet. The length of the sign is one foot more than the width. Find the length and width of the sign.

How did it go?

A rectangular garden has an area $15$ square feet. The length of the garden is two feet more than the width. Find the length and width of the garden.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

A rectangular sign has an area of 30 square feet. The length of the sign is one foot more than the width. Find the length and width of the sign.

How did it go?

Step 1. Read the problem. In problems involving geometric figures, a sketch can help you visualize the situation.
Step 2. Identify what you are looking for.	We are looking for the length and width.
Step 3. Name what you are looking for. The length is two feet more than width.	Let W = the width of the garden. W + 2 = the length of the garden
Step 4. Translate into an equation. Restate the important information in a sentence.	The area of the rectangular garden is 15 square feet.
Use the formula for the area of a rectangle.	$A = L \cdot W$
Substitute in the variables.	$15 = (W + 2) W$
Step 5. Solve the equation. Distribute first.	$15 = W^{2} + 2 W$
Get zero on one side.	$0 = W^{2} + 2 W - 15$
Factor the trinomial.	$0 = (W + 5) (W - 3)$
Use the Zero Product Property.	$0 = W + 5$	$0 = W - 3$
Solve each equation.	$−5 = W$	$3 = W$
Since W is the width of the garden, it does not make sense for it to be negative. We eliminate that value for W.	$−5 = W$ $W = 3$	$3 = W$ Width is 3 feet.
Find the value of the length.	$W + 2 = length$
	$3 + 2$
	$5$	Length is 5 feet.
Step 6. Check the answer. Does the answer make sense?

	Yes, this makes sense.
Step 7. Answer the question.	The width of the garden is 3 feet and the length is 5 feet.

Step 1. Read the problem. In problems involving geometric figures, a sketch can help you visualize the situation.
Step 2. Identify what you are looking for.	We are looking for the length and width.
Step 3. Name what you are looking for. The length is two feet more than width.	Let W = the width of the garden. W + 2 = the length of the garden
Step 4. Translate into an equation. Restate the important information in a sentence.	The area of the rectangular garden is 15 square feet.
Use the formula for the area of a rectangle.	$A = L \cdot W$
Substitute in the variables.	$15 = (W + 2) W$
Step 5. Solve the equation. Distribute first.	$15 = W^{2} + 2 W$
Get zero on one side.	$0 = W^{2} + 2 W - 15$
Factor the trinomial.	$0 = (W + 5) (W - 3)$
Use the Zero Product Property.	$0 = W + 5$	$0 = W - 3$
Solve each equation.	$−5 = W$	$3 = W$
Since W is the width of the garden, it does not make sense for it to be negative. We eliminate that value for W.	$−5 = W$ $W = 3$	$3 = W$ Width is 3 feet.
Find the value of the length.	$W + 2 = length$
	$3 + 2$
	$5$	Length is 5 feet.
Step 6. Check the answer. Does the answer make sense?

	Yes, this makes sense.
Step 7. Answer the question.	The width of the garden is 3 feet and the length is 5 feet.