Question

Find the missing side that the perimeter of the rectangle is 8x^2 + 2x + 8.

3x^2 - 2x + 5

a. 2x^2 + 6x - 2
b. 5x^2 + 13
c. 6x^2 - 4x + 10
d. x^2 + 3x - 1

Answer 1

To find the missing side of the rectangle given the perimeter is \(8x^2 + 2x + 8\) and one side is \(3x^2 - 2x + 5\), we can use the formula for the perimeter of a rectangle:

\[
P = 2(l + w)
\]

Where \(l\) is the length and \(w\) is the width. We can rearrange this formula to find the sum of the lengths:

\[
l + w = \frac{P}{2}
\]

Substituting the perimeter:

\[
l + w = \frac{8x^2 + 2x + 8}{2} = 4x^2 + x + 4
\]

Now we have:

\[
l + w = 4x^2 + x + 4
\]

Given that one side (let's assume length) \(l = 3x^2 - 2x + 5\), we can find the other side \(w\) by rearranging:

\[
w = (4x^2 + x + 4) - (3x^2 - 2x + 5)
\]

Now we can perform the subtraction:

\[
w = (4x^2 + x + 4) - (3x^2 - 2x + 5) = 4x^2 + x + 4 - 3x^2 + 2x - 5
\]

Combine like terms:

\[
= (4x^2 - 3x^2) + (x + 2x) + (4 - 5)
\]
\[
= x^2 + 3x - 1
\]

So, the missing side \(w\) is:

\[
w = x^2 + 3x - 1
\]

Looking at the answer choices, the correct choice is:

**d. \(x^2 + 3x - 1\)**.