Absolutely, let's go through each part in detail:
### Part A: Factoring the Area Expression of a Square
The given area expression of the square is:
[tex]\[ 16a^2 - 24a + 9 \][/tex]
We want to determine the side length of the square by factoring this expression completely.
1. Firstly, recognize that the expression is a perfect square trinomial. A perfect square trinomial has the form [tex]\((A - B)^2 = A^2 - 2AB + B^2\)[/tex].
2. We compare [tex]\(16a^2 - 24a + 9\)[/tex] to the perfect square trinomial form [tex]\((A - B)^2\)[/tex]:
- [tex]\(A^2 = 16a^2\)[/tex] so [tex]\(A = 4a\)[/tex]
- [tex]\(B^2 = 9\)[/tex] so [tex]\(B = 3\)[/tex]
- Check if the middle term fits: [tex]\(-2AB = -2 \cdot 4a \cdot 3 = -24a\)[/tex]
This fits perfectly with our original middle term, confirming it as a perfect square trinomial.
3. Therefore, the factored form of [tex]\(16a^2 - 24a + 9\)[/tex] is:
[tex]\[ (4a - 3)^2 \][/tex]
So, the length of each side of the square is:
[tex]\[ 4a - 3 \][/tex]
### Part B: Factoring the Area Expression of a Rectangle
The given area expression of the rectangle is:
[tex]\[ 9a^2 - 25b^2 \][/tex]
This expression is a difference of squares. Recall that the difference of squares can be factored using the identity:
[tex]\[ A^2 - B^2 = (A - B)(A + B)\][/tex]
1. Write the expression in the form [tex]\(A^2 - B^2\)[/tex]:
- [tex]\(A^2 = 9a^2\)[/tex] so [tex]\(A = 3a\)[/tex]
- [tex]\(B^2 = 25b^2\)[/tex] so [tex]\(B = 5b\)[/tex]
2. Apply the difference of squares formula:
[tex]\[ 9a^2 - 25b^2 = (3a)^2 - (5b)^2 = (3a - 5b)(3a + 5b) \][/tex]
Thus, the dimensions of the rectangle are:
[tex]\[ 3a - 5b \][/tex]
and
[tex]\[ 3a + 5b \][/tex]
### Final Summary
- For Part A, the length of each side of the square is [tex]\(4a - 3\)[/tex].
- For Part B, the dimensions of the rectangle are [tex]\(3a - 5b\)[/tex] and [tex]\(3a + 5b\)[/tex].
These factorizations provide the needed forms to describe the side of the square and the dimensions of the rectangle.
