8*x*(*x* – 9)1/2 + 4(*x* – 9)3/2 = 0

The given equation is:

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

To solve this equation, let's follow these steps:

### Step 1: Factor the common term

Notice that both terms on the left-hand side have a common factor of \( 4(x - 9)^{1/2} \). Let's factor this out:

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

### Step 2: Simplify the expression inside the parentheses

Simplify the expression inside the parentheses:

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

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

Factor out the 3 from the expression \( 3x - 9 \):

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

### Step 3: Set each factor to zero

Now, set each factor equal to zero:

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

For the first factor:

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

Square both sides:

\[
x - 9 = 0
\]

\[
x = 9
\]

For the second factor:

\[
x - 3 = 0
\]

\[
x = 3
\]

### Step 4: Check both solutions

Substitute \( x = 9 \) and \( x = 3 \) back into the original equation to verify if they satisfy the equation.

1. **For \( x = 9 \):**

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

   Simplifies to:

   \[
   0 + 0 = 0 \quad \text{(True)}
   \]

2. **For \( x = 3 \):**

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

   However, \( (3 - 9) = -6 \), and taking the square root or cube root of a negative number in the real number system results in complex numbers. So, \( x = 3 \) does not yield a real solution.

### Conclusion

The only real solution is:

\[
\boxed{x = 9}
\]

Would you like more details on any part of this solution?

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Here are five questions you might want to explore next:

1. How do you solve equations involving square roots?
2. What are the properties of radical expressions?
3. What happens when solving equations with no real solutions?
4. How can we simplify expressions before solving equations?
5. What are some examples of equations leading to complex solutions?

**Tip:** When dealing with equations involving square roots or higher powers, always check if the solutions are real or complex by considering the sign under the root.

Explore how to solve the equation 8x(x - 9)^(1/2) + 4(x - 9)^(3/2) = 0 step-by-step. Understand factorization and simplification of radical expressions involved in solving this problem. Suitable for advanced high school students.

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