The length of a rectangular garden is five feet more than twice the width. The area of the garden is 102 square feet. Find the length and width of the garden.

Answer 1

Translate these English sentences into math sentences:

Sentence 1: L = 5 + 2W

Sentence 2: A = 102

Now use the formula: A = L * W

Substitute sentence 1 in place of L in the area formula:

A = (5 + 2W) * W = 5W + 2W²

We now have the quadratic equation: 102 = 2W² + 5W OR:

2W² + 5W - 102 = 0

Solve by using quadratic formula:

W = (-5 +/- sqrt(5² - 4(2)(-102)) / 2(2)

W = (-5 +/- sqrt(25 + 816) / 4

W = (-5 +/- 29) / 4

so W = -34/4 or 24/4

The negative answer makes no sense for a width dimension, so the width is 6 feet.

Substitute back into sentence 1 to get the length 17 feet.

Check the answer: 17 ft * 6 ft = 102 square ft

NOTE: Since the quadratic formula worked out "nicely", we may notice in retrospect that we could have factored the quadratic equation:

2W² + 5W - 102 = (2W + 17)(W - 6)

From here, seeing that W = 6 or -17/2 is perhaps "easier" than doing the quadratic formula, but only if you saw that tricky quadratic factorization in the first place. It's good to become comfortable with the quadratic formula because it ALWAYS works (as does completing the square), whereas factoring only works if the quadratic coefficients are chosen "nicely."