solving a word problem using a quadratic equation with rational roots

Question

The length of a rectangle is
3 ft
less than double the width, and the area of the rectangle is
27 ft^2
Find the dimensions of the rectangle.

 

 

Answer=

Length: ___ ft

Width: ____ ft 

Answer 1

From the information given you can derive the equation.

l = 2w-3 and A= l x w = 27 ft²

27 ft2 = (2w-3) x w               substitute with value of l in terms of w

2w² -3w -27 = 0                   standard quadratic equation form ax² + bx + c = 0

 

the roots are:

w = [- b +/- sqr(b²-4ac)] / 2a

w = [- (-3) +/- sqr((-3)²-4(2)(-27))] / 2(2)

w = [3 +/- sqr(9 +216)] / 4

w = [3 +/- sqr(225)] / 4

w = [3 +/- 15] / 4

w = 18 / 4 and w = 12 / 4 = 3 

solve for l

l = 2w - 3 = 2(18/4) - 3 = 6 or l = 2w - 3 = 2(3) - 3 = 3

the two roots are 18/4 and 3

verify

l x w = 27

6 x 18/4 = 27 

27 = 27  so the root w=18/4 is valid

l x w = 27
3 x 3 ≠ 27
9 ≠ 27 so the root w=3 is not valid