To solve the equation:

\[
\sqrt{11 - 10x} = x - 2
\]

we need to square both sides to eliminate the square root:

\[
(\sqrt{11 - 10x})^2 = (x - 2)^2
\]

Simplifying, we get:

\[
11 - 10x = (x - 2)^2
\]

Expanding the right-hand side:

\[
11 - 10x = x^2 - 4x + 4
\]

Now, let's bring everything to one side to form a quadratic equation:

\[
x^2 + 6x - 7 = 0
\]

This is a standard quadratic equation, and we can solve it using the quadratic formula:

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

where \( a = 1 \), \( b = 6 \), and \( c = -7 \):

\[
x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 1 \cdot (-7)}}{2 \cdot 1}
\]

\[
x = \frac{-6 \pm \sqrt{36 + 28}}{2}
\]

\[
x = \frac{-6 \pm \sqrt{64}}{2}
\]

\[
x = \frac{-6 \pm 8}{2}
\]

Now, calculate the two possible values of \( x \):

1. \( x = \frac{-6 + 8}{2} = \frac{2}{2} = 1 \)
2. \( x = \frac{-6 - 8}{2} = \frac{-14}{2} = -7 \)

### Verification of Solutions:
We need to verify these solutions because squaring both sides might introduce extraneous solutions:

1. **Check \( x = 1 \):**
   \[
   \sqrt{11 - 10 \cdot 1} = 1 - 2
   \]
   \[
   \sqrt{1} = -1 \quad \text{(False)}
   \]

2. **Check \( x = -7 \):**
   \[
   \sqrt{11 - 10 \cdot (-7)} = -7 - 2
   \]
   \[
   \sqrt{81} = -9 \quad \text{(False)}
   \]

### Conclusion:
Both potential solutions, \( x = 1 \) and \( x = -7 \), fail upon verification. Therefore, **there is no solution** to the equation \(\sqrt{11 - 10x} = x - 2\).

Would you like further details or have any questions about this process?

Here are 5 related questions:
1. How do we identify extraneous solutions in equations involving square roots?
2. What happens when we square both sides of an equation?
3. How can we check the domain restrictions of square root functions?
4. Why do quadratic equations sometimes produce extraneous solutions?
5. How can graphing the functions help verify the solution?

**Tip:** Always verify your solutions when squaring both sides of an equation, as this can introduce solutions that don't satisfy the original equation.

Learn how to solve the square root equation sqrt(11 - 10x) = (x - 2) by squaring both sides and forming a quadratic equation. This step-by-step solution covers algebraic concepts, quadratic equations, and the importance of verifying extraneous solutions. Suitable for students in Grades 9-12.

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