Certainly! Let's work through each part step by step.
### Part A: Factoring the Area Expression of the Square
The given area of the square is [tex]\(4a^2 - 20a + 25\)[/tex] square units. To determine the length of each side, we need to factor the quadratic expression completely.
1. Identify the expression to factor: [tex]\(4a^2 - 20a + 25\)[/tex].
2. Check if it is a perfect square trinomial: A perfect square trinomial takes the form [tex]\((ma + b)^2 = m^2a^2 + 2mab + b^2\)[/tex].
Let's rewrite [tex]\(4a^2 - 20a + 25\)[/tex]:
- The first term [tex]\(4a^2\)[/tex] can be written as [tex]\((2a)^2\)[/tex].
- The last term [tex]\(25\)[/tex] can be written as [tex]\(5^2\)[/tex].
3. Check the middle term: The middle term should be [tex]\(2 \times 2a \times 5\)[/tex] which is [tex]\(20a\)[/tex]. Since we have [tex]\(-20a\)[/tex] in the expression, it verifies:
[tex]\[
(2a - 5)^2 = 4a^2 - 20a + 25
\][/tex]
4. Factor the expression:
[tex]\[
4a^2 - 20a + 25 = (2a - 5)^2
\][/tex]
5. Determine the length of each side of the square:
Since [tex]\((2a - 5)^2\)[/tex] represents the area of the square, the length of each side is:
[tex]\[
\boxed{2a - 5}
\][/tex]
### Part B: Factoring the Area Expression of the Rectangle
The given area of the rectangle is [tex]\(9a^2 - 16b^2\)[/tex] square units. To determine the dimensions of the rectangle, we need to factor the expression completely.
1. Identify the expression to factor: [tex]\(9a^2 - 16b^2\)[/tex].
2. Recognize the form of the expression: This is a difference of squares which takes the form [tex]\(m^2 - n^2 = (m - n)(m + n)\)[/tex].
Let's rewrite [tex]\(9a^2 - 16b^2\)[/tex]:
- The first term [tex]\(9a^2\)[/tex] can be written as [tex]\((3a)^2\)[/tex].
- The second term [tex]\(16b^2\)[/tex] can be written as [tex]\((4b)^2\)[/tex].
3. Apply the difference of squares formula:
[tex]\[
9a^2 - 16b^2 = (3a)^2 - (4b)^2
\][/tex]
[tex]\[
= (3a - 4b)(3a + 4b)
\][/tex]
4. Determine the dimensions of the rectangle:
The dimensions of the rectangle are given by the factors:
[tex]\[
\boxed{(3a - 4b)} \quad \text{and} \quad \boxed{(3a + 4b)}
\][/tex]
So, the factored dimensions of the rectangle are [tex]\(3a - 4b\)[/tex] and [tex]\(3a + 4b\)[/tex].
