You are planning a rectangular patio with length that is #7# #ft# less than three times its width. The area of the patio is #120# #ft^2#. What are the dimensions of the patio?

Since the length is defined by the width, let #x# represent the width. This means that the length will be represented by the expression #3x - 7#. The area of a rectangle is found by multiplying its length by its width. Substitute and solve for #w#.

#A = lw#
#120 = (3x - 7)x#
#120 = 3x^2 - 7x#
#120 - 120 = 3x^2 - 7x - 120#
#0 = 3x^2 - 7x - 120#

Now that the equation is simplified and in standard form, use the Quadratic Formula to find the possible solutions for #x#. The Quadratic Formula is #x = (-b +- sqrt(b^2 - 4ac))/(2a)#. For this situation, #a = 3#, #b = -7#, and #c = -120#.

#x = (-(-7) +- sqrt((-7)^2 - 4(3)(-120)))/(2*3)#
#x = (7 +- sqrt(49 +1440))/6#
#x = (7 +- sqrt(1489))/6#
#x ~~ (7+-39)/6#
#x ~~ (7 + 39)/6# or #x ~~ (7 - 39)/6#
#x ~~ 46/6# or #x ~~ -32/6#
#x ~~ 7.67# or #x ~~ -5.33#

Since distances (width, in this case) are not negative, disregard #-5.33#. The width of the rectangular patio is approximately #7.67#. Use this value to find the length of the patio.

#l = 3(7.67) - 7#
#l = 23.01 - 7#
#l = 16.01#

The dimensions of the rectangular patio are width = #7.67# ft and length = #16.01# ft.