Question
In front of a school are several gardens in rectangular raised beds. For each of the areas of a rectangular garden given, use factoring to find possible dimensions. Could the garden be square? If so, explain why.
a. x^2+32x+256
b. x^2-4y^2
c. x^2-20x+100
a. x^2+32x+256
b. x^2-4y^2
c. x^2-20x+100
Answers
GPT-4o mini
To find the possible dimensions of the rectangular gardens given the areas, we will factor each quadratic expression in the problems. We will also check if any of the gardens could be square by determining if there are equal factors.
### a. \(x^2 + 32x + 256\)
To factor \(x^2 + 32x + 256\), we look for two numbers that multiply to 256 and add up to 32.
The expression can be factored as follows:
\[
x^2 + 32x + 256 = (x + 16)(x + 16) = (x + 16)^2
\]
This means the dimensions of the garden could be \(x + 16\) by \(x + 16\).
Since both dimensions are equal, the garden **could** be square.
### b. \(x^2 - 4y^2\)
This expression is a difference of squares, which can be factored using the identity \(a^2 - b^2 = (a - b)(a + b)\):
\[
x^2 - 4y^2 = (x - 2y)(x + 2y)
\]
Thus, the possible dimensions of the garden could be \(x - 2y\) and \(x + 2y\).
For it to be a square, we would need \(x - 2y = x + 2y\), which implies:
\[
-2y = 2y \implies y = 0
\]
Since \(y = 0\) is not a generally acceptable condition (assuming \(y\) represents a length), in general for non-zero \(y\), the garden **cannot** be square.
### c. \(x^2 - 20x + 100\)
To factor \(x^2 - 20x + 100\), we look for two numbers that multiply to 100 and add to -20.
The expression can be factored as:
\[
x^2 - 20x + 100 = (x - 10)(x - 10) = (x - 10)^2
\]
Thus, the dimensions of the garden could be \(x - 10\) by \(x - 10\).
Since both dimensions are equal, the garden **could** be square.
### Summary:
- **(a)** Dimensions: \(x + 16\) by \(x + 16\) (could be square)
- **(b)** Dimensions: \(x - 2y\) by \(x + 2y\) (not square in general)
- **(c)** Dimensions: \(x - 10\) by \(x - 10\) (could be square)
### a. \(x^2 + 32x + 256\)
To factor \(x^2 + 32x + 256\), we look for two numbers that multiply to 256 and add up to 32.
The expression can be factored as follows:
\[
x^2 + 32x + 256 = (x + 16)(x + 16) = (x + 16)^2
\]
This means the dimensions of the garden could be \(x + 16\) by \(x + 16\).
Since both dimensions are equal, the garden **could** be square.
### b. \(x^2 - 4y^2\)
This expression is a difference of squares, which can be factored using the identity \(a^2 - b^2 = (a - b)(a + b)\):
\[
x^2 - 4y^2 = (x - 2y)(x + 2y)
\]
Thus, the possible dimensions of the garden could be \(x - 2y\) and \(x + 2y\).
For it to be a square, we would need \(x - 2y = x + 2y\), which implies:
\[
-2y = 2y \implies y = 0
\]
Since \(y = 0\) is not a generally acceptable condition (assuming \(y\) represents a length), in general for non-zero \(y\), the garden **cannot** be square.
### c. \(x^2 - 20x + 100\)
To factor \(x^2 - 20x + 100\), we look for two numbers that multiply to 100 and add to -20.
The expression can be factored as:
\[
x^2 - 20x + 100 = (x - 10)(x - 10) = (x - 10)^2
\]
Thus, the dimensions of the garden could be \(x - 10\) by \(x - 10\).
Since both dimensions are equal, the garden **could** be square.
### Summary:
- **(a)** Dimensions: \(x + 16\) by \(x + 16\) (could be square)
- **(b)** Dimensions: \(x - 2y\) by \(x + 2y\) (not square in general)
- **(c)** Dimensions: \(x - 10\) by \(x - 10\) (could be square)