Question

question 2 (essay worth 10 points) (07.03, 07.04 mc) part a: the area of a square is (4a² - 20a + 25) square units. determine the length of each side of the square by factoring the area expression completely. show your work. (5 points) part b: the area of a rectangle is (9a² - 16b²) square units. determine the dimensions of the rectangle by factoring the area expression completely. show your work. (5 points)

Explanation:

Step1: Factor the square's area expression

We have the expression $4a^{2}-20a + 25$. Notice that it is in the form of a perfect - square trinomial $x^{2}-2xy + y^{2}=(x - y)^{2}$, where $x = 2a$ and $y = 5$.
\[4a^{2}-20a + 25=(2a)^{2}-2\times(2a)\times5+5^{2}=(2a - 5)^{2}\]
Since the area of a square $A = s^{2}$ (where $s$ is the side - length), the side - length of the square is $(2a - 5)$ units.

Step2: Factor the rectangle's area expression

We have the expression $9a^{2}-16b^{2}$, which is a difference of squares. The formula for the difference of squares is $x^{2}-y^{2}=(x + y)(x - y)$. Here, $x = 3a$ and $y = 4b$.
\[9a^{2}-16b^{2}=(3a)^{2}-(4b)^{2}=(3a + 4b)(3a - 4b)\]
Since the area of a rectangle $A= l\times w$ (where $l$ is the length and $w$ is the width), the dimensions of the rectangle are $(3a + 4b)$ units and $(3a - 4b)$ units.

Answer:

Part A: The side - length of the square is $(2a - 5)$ units.
Part B: The dimensions of the rectangle are $(3a + 4b)$ units and $(3a - 4b)$ units.