The given equation is:

\[
\frac{x^3}{2x - 6} = x - 8
\]

### Step 1: Simplify the denominator
We can factor the denominator \( 2x - 6 \) as:

\[
2x - 6 = 2(x - 3)
\]

Thus, the equation becomes:

\[
\frac{x^3}{2(x - 3)} = x - 8
\]

### Step 2: Eliminate the denominator
To eliminate the denominator, multiply both sides of the equation by \( 2(x - 3) \):

\[
x^3 = (x - 8) \cdot 2(x - 3)
\]

### Step 3: Expand the right-hand side
First, expand \( (x - 8)(x - 3) \):

\[
(x - 8)(x - 3) = x^2 - 3x - 8x + 24 = x^2 - 11x + 24
\]

Now multiply this result by 2:

\[
2(x^2 - 11x + 24) = 2x^2 - 22x + 48
\]

Thus, the equation becomes:

\[
x^3 = 2x^2 - 22x + 48
\]

### Step 4: Rearrange the equation
Move all terms to one side:

\[
x^3 - 2x^2 + 22x - 48 = 0
\]

### Step 5: Solve the cubic equation
Now we need to solve the cubic equation:

\[
x^3 - 2x^2 + 22x - 48 = 0
\]

Let's check for possible rational roots using the Rational Root Theorem. The possible rational roots are the factors of \(-48\) divided by the factors of 1 (the leading coefficient of \(x^3\)). The possible roots are:

\[
\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 16, \pm 24, \pm 48
\]

Testing \(x = 2\):

\[
2^3 - 2(2^2) + 22(2) - 48 = 8 - 8 + 44 - 48 = 0
\]

So, \(x = 2\) is a root.

### Step 6: Factor the cubic equation
Since \(x = 2\) is a root, \(x - 2\) is a factor. We can divide \(x^3 - 2x^2 + 22x - 48\) by \(x - 2\) using synthetic or polynomial division.

After dividing, the quotient is \(x^2 + 0x + 24\), so:

\[
x^3 - 2x^2 + 22x - 48 = (x - 2)(x^2 + 24)
\]

### Step 7: Solve the remaining quadratic equation
The quadratic \(x^2 + 24 = 0\) has no real solutions because:

\[
x^2 = -24
\]

Thus, the solutions are imaginary:

\[
x = \pm \sqrt{-24} = \pm 2i\sqrt{6}
\]

### Final Solution:
The solutions are:

\[
x = 2, \quad x = 2i\sqrt{6}, \quad x = -2i\sqrt{6}
\]

Would you like more details or have any questions?

Here are five related questions for further practice:
1. How do you solve a cubic equation by factoring?
2. What is the Rational Root Theorem and how is it used?
3. How do you perform synthetic division for polynomial division?
4. What are imaginary numbers, and how do they arise in quadratic equations?
5. How can you simplify and factor a complex quadratic expression?

**Tip:** When solving higher-degree equations, always check for possible rational roots first to simplify the equation.

Learn how to solve the cubic equation x^3/(2(x - 3)) = x - 8 using algebraic methods like synthetic division and the Rational Root Theorem. This step-by-step solution covers key concepts such as factoring cubic equations and solving for imaginary numbers. Suitable for Grades 10-12 students.

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