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The given equation is:

\[
\sqrt{-6x + 19} = x - 2
\]

To solve this equation and check for extraneous solutions, let's follow these steps:

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

\[
(\sqrt{-6x + 19})^2 = (x - 2)^2
\]

This simplifies to:

\[
-6x + 19 = (x - 2)(x - 2)
\]

Now expand the right-hand side:

\[
-6x + 19 = x^2 - 4x + 4
\]

### Step 2: Rearrange the equation
Move all terms to one side:

\[
0 = x^2 - 4x + 4 + 6x - 19
\]

Simplify:

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

### Step 3: Factor the quadratic equation
Factor \(x^2 + 2x - 15\):

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

So, the solutions are:

\[
x = 3 \quad \text{or} \quad x = -5
\]

### Step 4: Check for extraneous solutions
We need to check both solutions by substituting them back into the original equation.

#### For \(x = 3\):

\[
\sqrt{-6(3) + 19} = 3 - 2
\]

Simplifying:

\[
\sqrt{-18 + 19} = 1 \quad \Rightarrow \quad \sqrt{1} = 1
\]

This is true, so \(x = 3\) is a valid solution.

#### For \(x = -5\):

\[
\sqrt{-6(-5) + 19} = -5 - 2
\]

Simplifying:

\[
\sqrt{30 + 19} = -7 \quad \Rightarrow \quad \sqrt{49} = -7
\]

This is false, since \(\sqrt{49} = 7\), not \(-7\).

Thus, \(x = -5\) is an extraneous solution.

### Final Answer:
The valid solution is \(x = 3\), so the correct answer is:

**C. \(x = 3\)**

Would you like more details or have any questions?

Here are five follow-up questions to expand on this topic:
1. What are extraneous solutions, and why do they occur?
2. How do you know when to check for extraneous solutions?
3. Can a square root ever produce a negative result?
4. How would you solve the same problem without squaring both sides first?
5. What are other methods of solving quadratic equations?

**Tip:** Always substitute your solutions back into the original equation when solving equations involving square roots to check for extraneous solutions.

Solve √(-6x + 19) = x - 2. Check for extraneous solutions.

This page provides a step-by-step solution to the equation √(-6x + 19) = x - 2, covering square roots, quadratic equations, and extraneous solutions. Learn how to check your solutions and avoid common errors. Suitable for students in Grades 10-12.

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