Question

Solve the radical equation. Check all proposed solutions.
StartRoot 15 minus 2 x EndRoot equals x
Question content area bottom
Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

Answer 1

To solve the radical equation \( \sqrt{15 - 2x} = x \), we will first square both sides to eliminate the square root:

\[
(\sqrt{15 - 2x})^2 = x^2
\]

This simplifies to:

\[
15 - 2x = x^2
\]

Rearranging the equation gives:

\[
x^2 + 2x - 15 = 0
\]

Next, we will factor the quadratic equation:

\[
(x + 5)(x - 3) = 0
\]

Setting each factor equal to zero gives us the possible solutions:

\[
x + 5 = 0 \quad \Rightarrow \quad x = -5
\]
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]

Now, we need to check each proposed solution in the original equation to verify if they are indeed solutions.

**Check \( x = -5 \)**:

Substituting \( x = -5 \) into the original equation:

\[
\sqrt{15 - 2(-5)} = -5
\]
\[
\sqrt{15 + 10} = -5
\]
\[
\sqrt{25} = -5
\]
\[
5 \neq -5 \quad (\text{not a solution})
\]

**Check \( x = 3 \)**:

Substituting \( x = 3 \) into the original equation:

\[
\sqrt{15 - 2(3)} = 3
\]
\[
\sqrt{15 - 6} = 3
\]
\[
\sqrt{9} = 3
\]
\[
3 = 3 \quad (\text{is a solution})
\]

After checking both proposed solutions, we find that the only valid solution to the equation \( \sqrt{15 - 2x} = x \) is:

\[
\boxed{3}
\]