quadratic? equation, quadratic formula.

Question

The length of a rectangular air filter is 5 inches less than twice the width. Find the length and width of the filter if the area is 273 square inches.

Answer 1

The information we are given look like this when we turn it into an equation.

 

A = L x W

L = 2W - 5

 

A = 273 in²

 

Seeing how L = a function of W we can replace L with the function of W.

 

273 = (2W - 5) x W

 

Distribute

 

273 = 2W² - 5W

 

and move everything to one side

 

0 = 2W² - 5W - 273

 

Yes, we use the quadratic formula at this point

 

(-b ± Sqr(b² - 4ac))/2a

 

a = 2  b = -5  c = -273

 

When we plug these in and calculate the quadratic formula we get

 

W = 13 and W = -21/2   we can ignore the -21/2 because a width can not be negative

 

So, our width is 13 inches

 

Now we take the width and plug it into the L equation we made earlier.

 

L = 2W - 5

L = 2(13) - 5 = 26 - 5 = 21 inches

L = 21 inches. To verify we will multiply the length and width together to find out if it gives the correct area

 

A = L x W

21 x 13 = 273in²

 

so W = 13 and L = 21 inches

 

Answer 2

L = 2W - 5

A  = 273 = WL = W(2W - 5)

 

273 = W(2W - 5)

 

273 = 2W² - 5W

 

2W² - 5W - 273 = 0

 

W = [1/(2*2)] {-(-5) ± √[(-5)² - 4(2)(-273)]}

 

W = (1/4)(5 ± 47)

 

W > 0, so use the positive square root

 

W = (5 + 47)/4 = 13

 

L = 2W - 5 = 2(13) - 5 = 21

 

width = 13 inches

length = 21 inches