To solve the equation:

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

### Step 1: Factor the equation.
First, we observe that both terms on the left-hand side have a common factor of \(4(x - 9)^{\frac{1}{2}}\). We can factor that out:

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

### Step 2: Simplify the expression inside the brackets.
Simplify the expression inside the brackets:

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

So the equation becomes:

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

### Step 3: Set each factor to zero.
Now, we have a product of two terms equal to zero. We can solve this by setting each factor to zero.

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

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

   Square both sides:

   \[
   x - 9 = 0
   \]

   So,

   \[
   x = 9
   \]

2. \(3x - 9 = 0\)

   Solve for \(x\):

   \[
   3x = 9
   \]

   \[
   x = 3
   \]

### Step 4: Verify the solutions.
Check both solutions in the original equation:

- For \(x = 9\):

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

  This simplifies to \(0 = 0\), so \(x = 9\) is a valid solution.

- For \(x = 3\):

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

  Since \((3 - 9)^{\frac{1}{2}}\) involves taking the square root of a negative number, which leads to a non-real (complex) number, \(x = 3\) is not a valid solution in the set of real numbers.

### Final Answer:
The solution to the equation is:

\[
\boxed{x = 9}
\]

Would you like more details or have any questions? Here are some related problems:

1. Solve \(2(x - 4)^{1/2} + (x - 4)^{3/2} = 0\).
2. Factor and solve \(5x(x - 2)^{1/3} + 3(x - 2)^{4/3} = 0\).
3. Solve \(x(x - 5)^{1/2} + 2(x - 5)^{3/2} = 0\).
4. Evaluate \(2(x - 7)^{1/2} + (x - 7)^{3/2} = 0\) at \(x = 7\).
5. Find the roots of \(x^2 - 6x + 9 = 0\).

**Tip:** Always check potential solutions by substituting them back into the original equation to ensure they work, especially when dealing with square roots or other even roots.

Learn how to solve the equation 8x(x – 9)^{1/2} + 4(x – 9)^{3/2} = 0 step-by-step. This problem involves equations with square roots and factorization techniques. Suitable for high school students.

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