Answer :
Certainly! Let's tackle each part of the question step-by-step.
### Part A: Determining the Side Length of the Square
The given area of the square is:
[tex]\[ 9a^2 - 24a + 16 \text{ square units} \][/tex]
First, we need to factor this quadratic expression completely.
#### Step 1: Recognize the quadratic form
The given expression [tex]\( 9a^2 - 24a + 16 \)[/tex] is a quadratic in the form of [tex]\( ax^2 + bx + c \)[/tex].
#### Step 2: Try to factor it as a perfect square trinomial
A perfect square trinomial takes the form [tex]\( (x - y)^2 = x^2 - 2xy + y^2 \)[/tex].
Let's attempt to write [tex]\( 9a^2 - 24a + 16 \)[/tex] in this form.
[tex]\[ 9a^2 - 24a + 16 = (3a - 4)^2 \][/tex]
We can verify by expanding [tex]\( (3a - 4)^2 \)[/tex]:
[tex]\[ (3a - 4)^2 = 9a^2 - 2 \cdot 3a \cdot 4 + 16 = 9a^2 - 24a + 16 \][/tex]
So, the quadratic expression [tex]\( 9a^2 - 24a + 16 \)[/tex] factors as [tex]\( (3a - 4)^2 \)[/tex].
#### Step 3: Determine the side length of the square
Since the area of the square is given by [tex]\( (3a - 4)^2 \)[/tex], the side length of the square is:
[tex]\[ 3a - 4 \][/tex]
Thus, each side of the square has a length of [tex]\( 3a - 4 \)[/tex] units.
### Part B: Determining the Dimensions of the Rectangle
The given area of the rectangle is:
[tex]\[ 25a^2 - 36b^2 \text{ square units} \][/tex]
#### Step 1: Recognize the quadratic form
The expression [tex]\( 25a^2 - 36b^2 \)[/tex] is a difference of squares. The difference of squares can be written as [tex]\( x^2 - y^2 = (x - y)(x + y) \)[/tex].
#### Step 2: Factor the difference of squares
We can factor the given expression by treating [tex]\( 25a^2 \)[/tex] and [tex]\( 36b^2 \)[/tex] as perfect squares.
[tex]\[ 25a^2 - 36b^2 = (5a)^2 - (6b)^2 \][/tex]
Using the difference of squares formula:
[tex]\[ (5a)^2 - (6b)^2 = (5a - 6b)(5a + 6b) \][/tex]
So, the expression [tex]\( 25a^2 - 36b^2 \)[/tex] factors as [tex]\( (5a - 6b)(5a + 6b) \)[/tex].
#### Step 3: Determine the dimensions of the rectangle
The factored form gives us the possible dimensions of the rectangle:
[tex]\[ (5a - 6b) \][/tex] and [tex]\[ (5a + 6b) \][/tex]
Thus, the dimensions of the rectangle are [tex]\( 5a - 6b \)[/tex] units and [tex]\( 5a + 6b \)[/tex] units.
### Summary:
- Part A: The side length of the square is [tex]\( 3a - 4 \)[/tex] units.
- Part B: The dimensions of the rectangle are [tex]\( 5a - 6b \)[/tex] units and [tex]\( 5a + 6b \)[/tex] units.
### Part A: Determining the Side Length of the Square
The given area of the square is:
[tex]\[ 9a^2 - 24a + 16 \text{ square units} \][/tex]
First, we need to factor this quadratic expression completely.
#### Step 1: Recognize the quadratic form
The given expression [tex]\( 9a^2 - 24a + 16 \)[/tex] is a quadratic in the form of [tex]\( ax^2 + bx + c \)[/tex].
#### Step 2: Try to factor it as a perfect square trinomial
A perfect square trinomial takes the form [tex]\( (x - y)^2 = x^2 - 2xy + y^2 \)[/tex].
Let's attempt to write [tex]\( 9a^2 - 24a + 16 \)[/tex] in this form.
[tex]\[ 9a^2 - 24a + 16 = (3a - 4)^2 \][/tex]
We can verify by expanding [tex]\( (3a - 4)^2 \)[/tex]:
[tex]\[ (3a - 4)^2 = 9a^2 - 2 \cdot 3a \cdot 4 + 16 = 9a^2 - 24a + 16 \][/tex]
So, the quadratic expression [tex]\( 9a^2 - 24a + 16 \)[/tex] factors as [tex]\( (3a - 4)^2 \)[/tex].
#### Step 3: Determine the side length of the square
Since the area of the square is given by [tex]\( (3a - 4)^2 \)[/tex], the side length of the square is:
[tex]\[ 3a - 4 \][/tex]
Thus, each side of the square has a length of [tex]\( 3a - 4 \)[/tex] units.
### Part B: Determining the Dimensions of the Rectangle
The given area of the rectangle is:
[tex]\[ 25a^2 - 36b^2 \text{ square units} \][/tex]
#### Step 1: Recognize the quadratic form
The expression [tex]\( 25a^2 - 36b^2 \)[/tex] is a difference of squares. The difference of squares can be written as [tex]\( x^2 - y^2 = (x - y)(x + y) \)[/tex].
#### Step 2: Factor the difference of squares
We can factor the given expression by treating [tex]\( 25a^2 \)[/tex] and [tex]\( 36b^2 \)[/tex] as perfect squares.
[tex]\[ 25a^2 - 36b^2 = (5a)^2 - (6b)^2 \][/tex]
Using the difference of squares formula:
[tex]\[ (5a)^2 - (6b)^2 = (5a - 6b)(5a + 6b) \][/tex]
So, the expression [tex]\( 25a^2 - 36b^2 \)[/tex] factors as [tex]\( (5a - 6b)(5a + 6b) \)[/tex].
#### Step 3: Determine the dimensions of the rectangle
The factored form gives us the possible dimensions of the rectangle:
[tex]\[ (5a - 6b) \][/tex] and [tex]\[ (5a + 6b) \][/tex]
Thus, the dimensions of the rectangle are [tex]\( 5a - 6b \)[/tex] units and [tex]\( 5a + 6b \)[/tex] units.
### Summary:
- Part A: The side length of the square is [tex]\( 3a - 4 \)[/tex] units.
- Part B: The dimensions of the rectangle are [tex]\( 5a - 6b \)[/tex] units and [tex]\( 5a + 6b \)[/tex] units.
