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:
- The perimeter \( P \) is given as \( 21x^2 - 5x + 4 \).
- One side (bottom of the shape) is \( 7x^2 + 3x + 2 \).
- The two sides are both \( 2x - 3 \).
- 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 \( 21x^2 - 5x + 4 \):
- \( 21x^2 - 7x^2 = 14x^2 \)
- \( -5x - 7x = -12x \)
- \( 4 - (-4) = 4 + 4 = 8 \)
Putting it all together, we have:
\[ s = 14x^2 - 12x + 8 \]
Thus, the length of the missing side is:
\[ \boxed{14x^2 - 12x + 8} \]
1 answer