8x(x – 9)1/2 + 4(x – 9)3/2 = 0 8x(x – 9)1/2 + 4(x – 9)3/2 = 0

To solve the equation:

\[
8x(x - 9)^{\frac{1}{2}} + 4(x - 9)^{\frac{3}{2}} = 0
\]

We can follow these steps:

1. **Factor the equation:**
   Factor out the common term \(4(x - 9)^{\frac{1}{2}}\):

   \[
   4(x - 9)^{\frac{1}{2}}[2x + (x - 9)] = 0
   \]

2. **Simplify the expression inside the brackets:**
   Simplify \(2x + (x - 9)\):

   \[
   2x + x - 9 = 3x - 9
   \]

   So the equation becomes:

   \[
   4(x - 9)^{\frac{1}{2}}(3x - 9) = 0
   \]

3. **Set each factor to zero:**
   The equation is satisfied if either factor equals zero:

   - \(4(x - 9)^{\frac{1}{2}} = 0\)
   - \(3x - 9 = 0\)

   **For the first factor**:
   \[
   (x - 9)^{\frac{1}{2}} = 0
   \]
   Square both sides:
   \[
   x - 9 = 0 \implies x = 9
   \]

   **For the second factor**:
   \[
   3x - 9 = 0
   \]
   Solve for \(x\):
   \[
   3x = 9 \implies x = 3
   \]

4. **Check both solutions in the original equation:**
   Substitute \(x = 9\) and \(x = 3\) back into the original equation to verify:

   - **For \(x = 9\)**:
     \[
     8(9)(9-9)^{\frac{1}{2}} + 4(9-9)^{\frac{3}{2}} = 0
     \]
     This is true since both terms involve \((0)^{\frac{1}{2}}\) or \((0)^{\frac{3}{2}}\).

   - **For \(x = 3\)**:
     \[
     8(3)(3-9)^{\frac{1}{2}} + 4(3-9)^{\frac{3}{2}} 
     \]
     The square root term \((3 - 9)^{\frac{1}{2}} = (-6)^{\frac{1}{2}}\) is undefined in the real number system.

   Thus, \(x = 3\) is not a valid solution.

### Final Answer:
The solution is \( \boxed{x = 9} \).

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How do you solve equations involving fractional exponents?
2. What are the properties of square roots and how do they affect solving equations?
3. How can you verify if an equation has imaginary or complex solutions?
4. What is the difference between solving equations in the real number system versus the complex number system?
5. How do you factor expressions with exponents?

**Tip:** Always check each solution by substituting it back into the original equation to ensure it satisfies all conditions.

This step-by-step guide helps you solve the radical equation 8x(x - 9)^(1/2) + 4(x - 9)^(3/2) = 0. The solution covers factoring, working with fractional exponents, and verifying solutions. Suitable for students in Grades 10-12.

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