For problems 8–10, each expression represents the area of a rectangle or square. Factor each expression to find the expressions that represent the length and width of each figure.

Question

(n^2 – 24n + 144) square feet

(9y^2 – 100) square meters

(x^2 + 7x – 18) square inches

Answer 1

There are some rules used here for factoring that you need to be able to recognize:

1.) a^{2 +} 2ab + b² = (a + b)² , and the subtraction version, a² - 2ab + b² = (a - b)² , “perfect square rule”

2.) a² - b² = (a -b )(a + b), “difference of squares rule”

3.)if the poly is in form: x² + mx + n, then find two numbers a,b so that a * b = n, and a + b = m.

(That is, the two numbers a,b multiply to the number(coefficient) at the end and add to the number in the middle.) This rule doesn’t have an official name, call it the “trinomial rule”.

You need to be able to recognize instances of these rules when they appear.

So for n² – 24n + 144, we have a square at the end just like the “perfect square rule”, 144 = 12². And also -24 = -2*12.

So n² – 24n + 144 = (n - 12)²

For the second one, 9y² – 100, notice only two terms and both are squares. This is an instance of the “difference of squares rule”. Then: 9y² – 100 = (3y -10)(3y + 10)

For the last one, x² + 7x – 18, we’ll use the “trinomial rule”: find two numbers a, b so that a * b = -18, and a + b = 7. The numbers that work for a, b are 9 and -2. Then x² + 7x – 18 = (x + 9)(x - 2).