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solving a word problem using a quadratic equation with rational roots

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 
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1 Answer

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