Solving a word problem using quadratic equation with rational roots

Question

the area of a rectangle is 50m^2, and the length of the rectangle is 5 m less than 3 times the width. Find the dimensions of the rectangle.

Answer 1

Area = length x width

The length = 3 times the width minus 5 so l = 3w - 5

Putting those together we get Area = (3w - 5)(w) or 50 = (3w - 5)(w)

Multiplying out the right side of the equation, we get 50 = 3w² - 5w then subtracting 50 from both sides we get:

3w² - 5w - 50 = 0

Factoring this quadratic we get:

(w - 5)(3w + 10) = 0

Setting each binomial equal to 0 we get:

w - 5 = 0

and 3w + 10 = 0

Solving these two equations we get w = 5 and w = -10/3

Obviously, the width cannot be -10/3 so we throw that solution out.

Plugging w = 5 into l = 3w - 5, we get l = 10

width = 5m and length = 10m

Answer 2

Area = Length · Width

Let x = the width. The length is 3x - 5:

50 = (3x-5)·x

50 = 3x² - 5x

0 = 3x² - 5x - 50

Solve by factoring or using the quadratic formula. You'll get two values for x, one positive and one negative. Since we can't have a negative width, the positive value is the answer.