20 Points WILL MARK BRAINLIEST IF YOU SHOW YOUR WORK
Part A: The area of a square is (16a2 − 24a + 9) 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 − 25b2) square units. Determine the dimensions of the rectangle by factoring the area expression completely. Show your work. (5 points)

Part A)

The length of each side of the square is (4a - 3).

Part B)

The dimensions of the rectangle are (3a + 5b) by (3a - 5b).

Step-by-step explanation:

Part A)

The area of a square is given by:

Notice that this resembles a perfect square trinomial, namely:

In fact, if we can see that:

Then we can let a = 4a and b = 3, and factor as such:

The area of a square is given by:

So:

Taking the square root of both sides produces:

So, the length of each side of the square is (4a - 3).

Remember that all four sides of a square are equivalent.

Part B)

The area of a rectangle is given by:

Again, notice that this resembles the form of the difference of two squares:

So, if we let a = 3a and b = 5b, we can see that:

So, we can factor accordingly:

The area of a rectangle is given:

So:

So, the dimensions of the rectangle are (3a + 5b) by (3a - 5b).

area of part 1:

a1=base*height/2=(8+2)*(8-2-2-1)/2=10*3/2=15 ft^2

area of part 2:

a2=length*width=(8+2)*1=10 ft2

area of part 3:

a3=length*width=8*2=16 ft2

total shaded area : a1+a2+a3=41 ft2

total area of the reactangle : a=16*8=128 ft2

total area of the nonshaded region : 128-41=87 ft2