To solve the problem, we need to determine the value of [tex]\( x \)[/tex] given the perimeter of the polygon [tex]\( ABCD \)[/tex]. We know the lengths of sides [tex]\( AB \)[/tex], [tex]\(\ BC \)[/tex], [tex]\( CD \)[/tex], and [tex]\( AD \)[/tex], and we know the total perimeter.
1. List the given information:
[tex]\[
AB = 12, \quad BC = \sqrt{2x + 54}, \quad CD = 10, \quad AD = 6
\][/tex]
2. Calculate the perimeter:
The perimeter of polygon [tex]\( ABCD \)[/tex] is given as 35. Hence, we can set up the following equation:
[tex]\[
AB + BC + CD + AD = 35
\][/tex]
3. Substitute the given lengths into the perimeter equation:
[tex]\[
12 + \sqrt{2x + 54} + 10 + 6 = 35
\][/tex]
4. Simplify the equation:
Combine the known lengths:
[tex]\[
12 + 10 + 6 = 28
\][/tex]
Thus, the equation becomes:
[tex]\[
28 + \sqrt{2x + 54} = 35
\][/tex]
5. Isolate the square root term:
Subtract 28 from both sides:
[tex]\[
\sqrt{2x + 54} = 35 - 28
\][/tex]
[tex]\[
\sqrt{2x + 54} = 7
\][/tex]
6. Square both sides to eliminate the square root:
[tex]\[
(\sqrt{2x + 54})^2 = 7^2
\][/tex]
[tex]\[
2x + 54 = 49
\][/tex]
7. Solve for [tex]\( x \)[/tex]:
Subtract 54 from both sides:
[tex]\[
2x = 49 - 54
\][/tex]
[tex]\[
2x = -5
\][/tex]
Divide by 2:
[tex]\[
x = \frac{-5}{2}
\][/tex]
Therefore, the radical equation to be solved is
[tex]\[
\sqrt{2x + 54} + 28 = 35
\][/tex]
and the solution to this equation is
[tex]\[
x = -\frac{5}{2}
\][/tex]
