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The equation provided is:

\[
\frac{3x}{x-2} = \frac{-18}{x^2 - 7x + 10}
\]

To solve for \(x\), follow these steps:

### Step 1: Cross-Multiply
Since the equation is a proportion, you can cross-multiply:

\[
3x \cdot (x^2 - 7x + 10) = -18 \cdot (x - 2)
\]

### Step 2: Expand both sides
Expand both sides of the equation:

\[
3x(x^2 - 7x + 10) = -18(x - 2)
\]

Expanding the left side:

\[
3x^3 - 21x^2 + 30x
\]

Expanding the right side:

\[
-18x + 36
\]

So, the equation becomes:

\[
3x^3 - 21x^2 + 30x = -18x + 36
\]

### Step 3: Bring all terms to one side of the equation
Move all terms to the left side to set the equation to zero:

\[
3x^3 - 21x^2 + 30x + 18x - 36 = 0
\]

Combine like terms:

\[
3x^3 - 21x^2 + 48x - 36 = 0
\]

### Step 4: Factor the equation
First, factor out the common factor of 3:

\[
3(x^3 - 7x^2 + 16x - 12) = 0
\]

So, you have:

\[
x^3 - 7x^2 + 16x - 12 = 0
\]

### Step 5: Find the roots
You can solve this cubic equation using the Rational Root Theorem, synthetic division, or a numerical method to find the exact roots.

After applying the Rational Root Theorem, we find that the roots of the cubic equation are \( x = 1 \), \( x = 2 \), and \( x = 6 \).

### Step 6: Check for any extraneous solutions
Substitute the roots back into the original equation to ensure they do not make any denominator zero. The potential denominators are \(x-2\) and \(x^2 - 7x + 10\).

Checking:

1. For \(x = 1\): Denominator \( x - 2 \) is not zero, and \(1^2 - 7(1) + 10 = 4\), so it's valid.
2. For \(x = 2\): The denominator \(x - 2\) becomes zero, so \(x = 2\) is **not** a valid solution.
3. For \(x = 6\): Denominator \( x - 2 \) is not zero, and \(6^2 - 7(6) + 10 = 4\), so it's valid.

Thus, the solutions are:

\[
\boxed{1, 6}
\]

Do you have any questions, or would you like further details on any part of the solution?

### Related Questions:
1. How do you apply the Rational Root Theorem to cubic equations?
2. What are some methods to solve cubic equations?
3. How do you determine if a solution is extraneous?
4. Why is it important to check for solutions that might make the denominator zero?
5. What is synthetic division, and how is it used in solving polynomial equations?

**Tip:** Always check for extraneous solutions in rational equations, as certain solutions might make the denominator zero and hence are not valid.

Solve for x: (3x) / (x-2) = -18 / (x^2 - 7x + 10)

Learn how to solve the rational equation (3x)/(x-2) = -18/(x^2 - 7x + 10) step by step. This solution covers cross-multiplication, factoring cubic equations, and identifying valid solutions while checking for extraneous ones. Suitable for students in Grades 10-12.

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