Sure! Let's go through each part step-by-step to solve the given problems.
### Part A: Factoring the Area of a Square
The area of the square is given by the expression [tex]\( 9x^2 - 12x + 4 \)[/tex]. To determine the length of each side, we need to factor this quadratic expression completely.
1. Identify the Quadratic Expression:
[tex]\( 9x^2 - 12x + 4 \)[/tex]
2. Check for Factorization as a Perfect Square:
We suspect this quadratic to be a perfect square trinomial because it matches the form [tex]\( (ax + b)^2 = a^2x^2 + 2abx + b^2 \)[/tex].
3. Compare and Factorize:
The quadratic term is [tex]\( 9x^2 \)[/tex], which is [tex]\( (3x)^2 \)[/tex].
The constant term is 4, which is [tex]\( 2^2 \)[/tex].
The middle term is [tex]\(-12x\)[/tex], which is twice the product of [tex]\( 3x \)[/tex] and [tex]\(-2\)[/tex] (i.e., [tex]\( 2(3x)(-2) = -12x \)[/tex]).
Thus, we can write [tex]\( 9x^2 - 12x + 4 \)[/tex] as:
[tex]\[ 9x^2 - 12x + 4 = (3x - 2)^2 \][/tex]
So, the length of each side of the square is [tex]\( 3x - 2 \)[/tex].
### Part B: Factoring the Area of a Rectangle
The area of the rectangle is given by the expression [tex]\( 25x^2 - 16y^2 \)[/tex]. We need to determine the dimensions of the rectangle by factoring this expression completely.
1. Identify the Expression:
[tex]\( 25x^2 - 16y^2 \)[/tex]
2. Recognize the Difference of Squares:
This expression is a difference of squares, which can be generally factored using the identity [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex].
3. Apply the Difference of Squares Formula:
Here, [tex]\( a^2 = (5x)^2 \)[/tex] and [tex]\( b^2 = (4y)^2 \)[/tex], so:
[tex]\[ 25x^2 - 16y^2 = (5x)^2 - (4y)^2 \][/tex]
[tex]\[ 25x^2 - 16y^2 = (5x - 4y)(5x + 4y) \][/tex]
Therefore, the dimensions of the rectangle are [tex]\( 5x - 4y \)[/tex] and [tex]\( 5x + 4y \)[/tex].
### Summary
- Part A: The length of each side of the square is [tex]\( 3x - 2 \)[/tex].
- Part B: The dimensions of the rectangle are [tex]\( 5x - 4y \)[/tex] and [tex]\( 5x + 4y \)[/tex].
These factorized forms of the given expressions provide the required side lengths and rectangle dimensions.
