The given equation is:

$x - 5 = \sqrt{12x + 25}$

Part 1a: Eliminate the Radical

To eliminate the square root, square both sides of the equation:

$(x - 5)^2 = (\sqrt{12x + 25})^2$

Simplifying both sides:

$(x - 5)^2 = 12x + 25$

Now, expand the left side:

$(x - 5)^2 = x^2 - 10x + 25$

So the equation becomes:

$x^2 - 10x + 25 = 12x + 25$

Now, let's check which option matches this form:

• Option A: $x^2 - 10x + 25 = 144x + 25$ → Incorrect.
• Option B: $x^2 - 10x + 25 = 12x + 25$ → Correct!
• Option C: $x^2 - 5 = 12x + 25$ → Incorrect.
• Option D: $x - 5 = 12x + 25$ → Incorrect.

The correct answer is Option B.

Would you like to proceed with solving the rest of the equation, or do you have any other questions?

Here are some related questions to explore:

1. How do you solve the equation after eliminating the radical?
2. How do you check for extraneous solutions in radical equations?
3. Can you solve other radical equations like $x - 3 = \sqrt{8x + 15}$?
4. What happens when the radical term is more complex, like $\sqrt{2x + 3}$?
5. What are the general steps to solve equations involving square roots?

Tip: Always check for extraneous solutions when solving radical equations, as squaring both sides can introduce false solutions.