Solve the radical equation. Complete parts a and b. 

x minus 5 equals StartRoot 12 x plus 25 EndRootx−5=12x+25

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Part 1

a. Write a new equation that does not contain a radical. Choose the correct answer.

A.

x squared minus 10 x plus 25x2−10x+25equals=144144xplus+2525

B.

x squared minus 10 x plus 25x2−10x+25equals=1212xplus+2525

C.

xsquared2minus−55equals=1212xplus+2525

D.

xminus−55equals=1212xplus+25

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.

Solve the radical equation. Complete parts a and b.
x - 5 = √(12x + 25)

Part 1
a. Write a new equation that does not contain a radical. Choose the correct answer:
A) x² - 10x + 25 = 144x + 25
B) x² - 10x + 25 = 12x + 25
C) x² - 5 = 12x + 25
D) x - 5 = 12x + 25

Learn how to solve the radical equation x - 5 = √(12x + 25) by eliminating the square root and solving the quadratic equation. This detailed guide walks through squaring both sides, expanding binomials, and solving for x, suitable for students in Grades 9-10.

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