Solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

Square root of (30 − x) = x

The given equation is:

\[
\sqrt{30 - x} = x
\]

### Step 1: Square both sides
To eliminate the square root, square both sides of the equation:

\[
(\sqrt{30 - x})^2 = x^2
\]

This simplifies to:

\[
30 - x = x^2
\]

### Step 2: Rearrange into a quadratic equation
Move all terms to one side:

\[
x^2 + x - 30 = 0
\]

### Step 3: Solve the quadratic equation
Now, solve the quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(x^2 + x - 30 = 0\), we have:
- \(a = 1\)
- \(b = 1\)
- \(c = -30\)

Plug these into the quadratic formula:

\[
x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-30)}}{2(1)}
\]

\[
x = \frac{-1 \pm \sqrt{1 + 120}}{2}
\]

\[
x = \frac{-1 \pm \sqrt{121}}{2}
\]

\[
x = \frac{-1 \pm 11}{2}
\]

Thus, the two possible solutions are:

\[
x = \frac{-1 + 11}{2} = \frac{10}{2} = 5
\]
\[
x = \frac{-1 - 11}{2} = \frac{-12}{2} = -6
\]

### Step 4: Check for extraneous solutions
Now, substitute both values back into the original equation to check for extraneous solutions.

For \(x = 5\):

\[
\sqrt{30 - 5} = 5
\]
\[
\sqrt{25} = 5 \quad \text{(True)}
\]

For \(x = -6\):

\[
\sqrt{30 - (-6)} = -6
\]
\[
\sqrt{36} = -6 \quad \text{(False, since \(\sqrt{36} = 6\))}
\]

Therefore, \(x = -6\) is an extraneous solution.

### Final Answer:
The only solution is:

\[
x = 5
\]

Would you like more details on any step? Here are some related questions to explore further:

1. How do you identify extraneous solutions in radical equations?
2. Can all radical equations have extraneous solutions?
3. What are common mistakes when solving equations with square roots?
4. How does the quadratic formula work, and why is it useful in this case?
5. What other methods can solve the quadratic equation besides the quadratic formula?

**Tip:** Always check your solutions in the original equation when solving radical equations to avoid extraneous solutions.

Solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) Square root of (30 − x) = x

Learn how to solve the radical equation √(30 - x) = x with a step-by-step solution. This problem involves squaring both sides, solving a quadratic equation, and checking for extraneous solutions. Suitable for students in Grades 9-12.

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