Question

Solve the radical equation. Check all proposed solutions.
StartRoot x plus 16 EndRoot minus StartRoot x minus 8 EndRoot equals 2

Answer 1

To solve the equation \( \sqrt{x + 16} - \sqrt{x - 8} = 2 \), we begin by isolating one of the square root terms.

Let's isolate \( \sqrt{x + 16} \):

\[
\sqrt{x + 16} = \sqrt{x - 8} + 2
\]

Next, we square both sides to eliminate the square roots:

\[
(\sqrt{x + 16})^2 = (\sqrt{x - 8} + 2)^2
\]

This gives us:

\[
x + 16 = (x - 8) + 4\sqrt{x - 8} + 4
\]

Simplifying the right side:

\[
x + 16 = x - 8 + 4\sqrt{x - 8} + 4
\]

\[
x + 16 = x - 4 + 4\sqrt{x - 8}
\]

Next, we can subtract \( x \) from both sides:

\[
16 = -4 + 4\sqrt{x - 8}
\]

Now, add 4 to both sides:

\[
20 = 4\sqrt{x - 8}
\]

Dividing by 4:

\[
5 = \sqrt{x - 8}
\]

Next, we square both sides again to eliminate the square root:

\[
25 = x - 8
\]

Adding 8 to both sides gives us:

\[
x = 33
\]

Now, we need to check if this solution satisfies the original equation:

Substituting \( x = 33 \) back into the original equation:

\[
\sqrt{33 + 16} - \sqrt{33 - 8} = \sqrt{49} - \sqrt{25} = 7 - 5 = 2
\]

Since both sides of the equation are equal, \( x = 33 \) is indeed a solution.

Thus, the final solution is:

\[
\boxed{33}
\]