The area of a rectangle is (16x2 − 9y2) square units. Determine the dimensions of the rectangle by factoring the area expression completely. Show your work.
We know that the area of a rectangle is given by the product of its length and width. Let the length of the rectangle be represented by 16x2 and its width by -9y2. This may initially seem counterintuitive due to the negative sign, but it is necessary for factoring later on.
Therefore, we have:
Area = length x width
(16x2) (-9y2) = -144x2 y2
To factor this expression, we can use the difference of squares formula:
a2 - b2 = (a + b)(a - b)
In our case, a = 4x and b = 3y, so:
144x2 y2 = (4x)2 - (3y)2
= (4x + 3y)(4x - 3y)
Therefore, the dimensions of the rectangle are:
Length = 4x + 3y
Width = 4x - 3y
Can you paraphrase your answer
To determine the dimensions of a rectangle given its area expression of (16x2 − 9y2) square units, we need to factor it completely. The product of length and width will give us the area of the rectangle. Assuming the length to be 16x2 and the width to be -9y2, we can express the area as (-144x2 y2). To factor this expression, we can use the difference of squares formula, which gives us the factors (4x + 3y) and (4x - 3y). Hence, the dimensions of the rectangle are Length = 4x + 3y and Width = 4x - 3y.
To determine the dimensions of the rectangle, we need to factor the area expression completely. The given area expression is (16x^2 - 9y^2) square units.
Let's start by factoring the expression (16x^2 - 9y^2). The given expression can be rewritten as the difference of squares:
16x^2 - 9y^2
This can be written as:
(4x)^2 - (3y)^2
Now, we have the difference of squares:
(a^2 - b^2) = (a + b)(a - b)
Applying this formula, we can factor the expression further:
(4x)^2 - (3y)^2 = (4x + 3y)(4x - 3y)
Therefore, the factored form of the area expression (16x^2 - 9y^2) is (4x + 3y)(4x - 3y).
To determine the dimensions of the rectangle, we equate the factors to find the length and width of the rectangle. So we have:
4x + 3y = length
4x - 3y = width
Therefore, the dimensions of the rectangle are (4x + 3y) units for the length and (4x - 3y) units for the width.