Question

solve the radical equation. check your solution(s). write
o solution\ if there is no solution. (sqrt{14 - 3x}=sqrt{2x + 9}) (x=square)

Explanation:

Step1: Square both sides to eliminate radicals

To solve the radical equation \(\sqrt{14 - 3x}=\sqrt{2x + 9}\), we start by squaring both sides. Squaring a square root cancels out the square root, so we get:
\[
(\sqrt{14 - 3x})^2=(\sqrt{2x + 9})^2
\]
Simplifying both sides, we have:
\[
14 - 3x=2x + 9
\]

Step2: Solve for \(x\)

Now, we solve the resulting linear equation for \(x\). First, we add \(3x\) to both sides to get the \(x\)-terms on one side:
\[
14=2x + 3x+ 9
\]
Combining like terms, we have:
\[
14 = 5x+9
\]
Next, we subtract 9 from both sides:
\[
14 - 9=5x
\]
Simplifying the left side gives:
\[
5 = 5x
\]
Then, we divide both sides by 5:
\[
x = 1
\]

Step3: Check the solution

We need to check if \(x = 1\) satisfies the original equation. Substitute \(x = 1\) into the original equation:
Left side: \(\sqrt{14-3(1)}=\sqrt{14 - 3}=\sqrt{11}\)
Right side: \(\sqrt{2(1)+9}=\sqrt{2 + 9}=\sqrt{11}\)
Since both sides are equal when \(x = 1\), the solution is valid.

Answer:

\(x = 1\)