Question

The perimeter of the shape is 21x^2 - 5x + 4. Find the length of the missing side. The bottom of the shape shows: 7x^2 + 3x + 2. The two sides the shape has shows: 2x - 3.

Answer 1

To find the length of the missing side of the shape, we can use the information provided about the perimeter. The perimeter \( P \) of the shape can be expressed as the sum of all its sides.

From the problem:

1. The perimeter \( P \) is given as \( 21x^2 - 5x + 4 \).
2. One side (bottom of the shape) is \( 7x^2 + 3x + 2 \).
3. The two sides are both \( 2x - 3 \).
4. We need to find the length of the missing side, which we'll call \( s \).

The formula for the perimeter of a polygon is the sum of its side lengths:

\[
P = \text{(bottom side)} + \text{(side 1)} + \text{(side 2)} + \text{(missing side)}
\]

Substituting what we know, we have:

\[
21x^2 - 5x + 4 = (7x^2 + 3x + 2) + (2x - 3) + (2x - 3) + s
\]

Now let's simplify the right-hand side. First, we can combine the two side lengths:

\[
(2x - 3) + (2x - 3) = 4x - 6
\]

So the equation now looks like:

\[
21x^2 - 5x + 4 = (7x^2 + 3x + 2) + (4x - 6) + s
\]

Let's combine the terms on the right-hand side.

Adding \( 7x^2 \), \( 3x \), and \( 4x \) gives:

\[
7x^2 + 3x + 4x = 7x^2 + 7x
\]

And combining the constant terms:

\[
2 - 6 = -4
\]

So we have:

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

Now let's isolate \( s \):

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

Now subtract \( 7x^2 + 7x - 4 \) from \( 21x^2 - 5x + 4 \):

1. \( 21x^2 - 7x^2 = 14x^2 \)
2. \( -5x - 7x = -12x \)
3. \( 4 - (-4) = 4 + 4 = 8 \)

Putting it all together, we have:

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

Thus, the length of the missing side is:

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