Question

the perimeter of the shape below is 21x^2 - 5x + 4 find the length of the missing side
4x^2 + 5
-14x^2 + 12x - 8
14x^2 - 12x - 8
14x^2 - 12x + 8
4x^2 - 8

Answer 1

To find the length of the missing side of the shape, we need to use the information given about the perimeter. The perimeter \( P \) of a shape is the sum of the lengths of all its sides.

Let \( P = 21x^2 - 5x + 4 \) represent the perimeter and assume we have a total of four sides, but we know the lengths of three of them. Let's denote the lengths of these sides as follows:

1. Side 1: \( 4x^2 + 5 \)
2. Side 2: \( -14x^2 + 12x - 8 \)
3. Side 3: \( 14x^2 - 12x - 8 \)

Let’s denote the missing side as \( S \). According to the perimeter formula:

\[
P = \text{Side 1} + \text{Side 2} + \text{Side 3} + S
\]

Substituting in the expressions for the sides we have:

\[
21x^2 - 5x + 4 = (4x^2 + 5) + (-14x^2 + 12x - 8) + (14x^2 - 12x - 8) + S
\]

Let’s simplify the right side:

1. Combine the quadratic terms:
\[
4x^2 - 14x^2 + 14x^2 = 4x^2
\]

2. Combine the linear terms:
\[
12x - 12x = 0
\]

3. Combine the constant terms:
\[
5 - 8 - 8 = -11
\]

So, the simplified equation becomes:
\[
21x^2 - 5x + 4 = 4x^2 - 11 + S
\]

Now, isolate \( S \) by moving everything else to the left side:
\[
S = (21x^2 - 5x + 4) - (4x^2 - 11)
\]

Distributing the negative sign:
\[
S = 21x^2 - 5x + 4 - 4x^2 + 11
\]

Combine like terms:
\[
S = (21x^2 - 4x^2) + (-5x) + (4 + 11) = 17x^2 - 5x + 15
\]

Thus, the length of the missing side is:
\[
\boxed{17x^2 - 5x + 15}
\]