QUADRATIC EAQUATION

Question

The area of a bulletin board is 52squarefeet. The length is three feet less than four times the width. Find the length and the width of the bulletin board.

Answer 1

We have two variables and two equations:

1. length * width = 52
2. length = 4 * width - 3

Therefore we can plug one equation into the other to find the length and width:

1. (4 * width - 3) * width = 52
2. 4 * width² - 3 * width - 52 = 0

Using quadratic formula (-b ± √(b²-4ac))/(2a):

1. width = (3 ±√(3²-4(4)(-52)))/(2 * 4)
2. width = (3 ± 29)/8

Therefore width is either 4' or -3.25', but we take the positive value for lengths, so width is 4'

Plugging this back into equation 2 gives:

1. length = 4 * (4) - 3

Therefore the length is 13'. Also plugging back into equation 1: 13 * 4 = 52 is satisfied

Answer 2

This type of problem is a frequent flyer on the questions board! To do this problem, first set up your variables, and identify the area equation for a rectangle; a bulletin board is usually a rectangle.

A = L * W (the * means "times")

A = area

L = length

W = width

Now, use the information in the problem to specify length in terms of width. Here's how:

You know the length (L) is 3 feet less than four times the width. Four times the width is 4W. You need to subtract 3 feet from this to get the length. So:

L = 4W - 3

Now, plug your information into the equation, using the given area = 52 square feet:

52 = (4W-3) * W

Multiply the right side out using distribution:

52 = 4W^2 - 3W (the ^2 means "squared")

Subtract 52 from both sides:

0 = 4W^2 - 3W - 52

I will let you finish this... it is now a factoring problem. Use any factoring method you like.

Note: You will get two answers for W. One answer will be negative and will therefore be invalid, because you cannot have negative lengths. Use the positive answer for W to substitute into the length equation (L = 4W - 3) to get length.

Check your answer to assure that when you multiply L * W you do get 52. ALWAYS CHECK YOUR ANSWER!! :-)