To solve the equation \((y - x)^3 = cx\), we can approach this step by step.

### Step 1: Expand the left-hand side
The equation is:
\[
(y - x)^3 = cx
\]
Expanding \((y - x)^3\) using the binomial theorem:
\[
(y - x)^3 = y^3 - 3y^2x + 3yx^2 - x^3
\]
So, the equation becomes:
\[
y^3 - 3y^2x + 3yx^2 - x^3 = cx
\]

### Step 2: Simplify and rearrange
Rearrange all terms to one side:
\[
y^3 - 3y^2x + 3yx^2 - x^3 - cx = 0
\]
This is a cubic equation in \( y \).

### Interpretation:
This is a complex cubic equation, and finding a general solution for \( y \) in terms of \( x \) and \( c \) may involve advanced techniques such as the cubic formula. Specific solutions might require numeric methods or assumptions about \( c \) or \( x \).

If you have more details or specific values for \( c \) and \( x \), I can help find a more precise solution.

Would you like a detailed solution or numerical example? Here are some related questions:

1. What happens if \( c = 0 \)?
2. How do we solve a cubic equation?
3. Can we factor the equation for specific values of \( c \)?
4. What if \( x = y \)?
5. How does the discriminant of a cubic equation work?

**Tip:** When dealing with polynomial equations, try factoring first to simplify the problem.

Learn how to solve the cubic equation (y - x)^3 = cx with a detailed step-by-step solution. This problem covers algebraic concepts such as cubic equations and the binomial theorem. Suitable for students in Grades 10-12.

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