Let's analyze the equation step by step.

Given equation:

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

Expand the left-hand side using the formula for the square of a binomial, \((a - b)^2 = a^2 - 2ab + b^2\):

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

Simplifying:

\[
x - 20\sqrt{x} + 100 = x + 10
\]

Now, subtract \(x\) from both sides:

\[
-20\sqrt{x} + 100 = 10
\]

Next, subtract 100 from both sides:

\[
-20\sqrt{x} = -90
\]

Now, divide both sides by \(-20\):

\[
\sqrt{x} = \frac{9}{2}
\]

Finally, square both sides to solve for \(x\):

\[
x = \left(\frac{9}{2}\right)^2 = \frac{81}{4}
\]

So, the solution is:

\[
x = \frac{81}{4} \quad \text{or} \quad x = 20.25
\]

To verify, substitute \(x = 20.25\) back into the original equation and check if both sides are equal. This confirms that \(x = 20.25\) is indeed a solution.

Would you like further details, or do you have any other questions?

Here are 5 related questions you might find interesting:
1. What is the binomial expansion, and how is it applied?
2. How do you solve equations involving square roots?
3. What are common mistakes in solving radical equations?
4. How can you verify the solution to an equation?
5. What are some applications of solving radical equations in real life?

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

Discover how to solve the radical equation (sqrt(x) - 10)^2 = x + 10 with a step-by-step explanation. This solution covers binomial expansion, properties of square roots, and solving radical equations. Ideal for students in Grades 9-11.

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