Question

the perimeter of the shape is 21x^2 - 5x + 4.
side 1: 2x - 3
side 2: 2x - 3
side 3: 7x^2 + 3x + 2
find the length of the missing side

Answer 1

To find the missing side of the shape given the perimeter and the lengths of the other three sides, we can use the perimeter formula:

\[
\text{Perimeter} = \text{Side 1} + \text{Side 2} + \text{Side 3} + \text{Missing Side}
\]

We know:
- Perimeter = \( 21x^2 - 5x + 4 \)
- Side 1 = \( 2x - 3 \)
- Side 2 = \( 2x - 3 \)
- Side 3 = \( 7x^2 + 3x + 2 \)

Let's denote the length of the missing side as \( \text{Missing Side} = M \).

First, calculate the total length of sides 1, 2, and 3:

\[
\text{Total length of known sides} = (2x - 3) + (2x - 3) + (7x^2 + 3x + 2
\]

Combining the known sides:

\[
= (2x + 2x + 3x + 7x^2) + (-3 - 3 + 2)
\]

Combining like terms:

\[
= 7x^2 + 7x - 4
\]

Now we set up the equation using the perimeter:

\[
21x^2 - 5x + 4 = (7x^2 + 7x - 4) + M
\]

To isolate \( M \), we rearrange the equation:

\[
M = (21x^2 - 5x + 4) - (7x^2 + 7x - 4)
\]

Subtracting the known sides from the perimeter:

\[
M = 21x^2 - 5x + 4 - 7x^2 - 7x + 4
\]

Combine like terms:

\[
M = (21x^2 - 7x^2) + (-5x - 7x) + (4 + 4)
\]

\[
M = 14x^2 - 12x + 8
\]

So, the missing side is:

\[
\boxed{14x^2 - 12x + 8}
\]