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Part A:

The area of a square is [tex]\left(16 a^2 - 24 a + 9\right)[/tex] 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 [tex]\left(9 a^2 - 25 b^2\right)[/tex] square units. Determine the dimensions of the rectangle by factoring the area expression completely. Show your work. (5 points)

Sagot :

GinnyAnsweridn
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.
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