x²/(x-1)=x³/(x-2)
x²(x-2)=x³(x-1)
x³-2x²=x⁴-x³
-x⁴+2x³-2x²=0

Let's solve the equation step by step.

We are given:

\[
\frac{x^2}{x-1} = \frac{x^3}{x-2}
\]

### Step 1: Cross-multiply
We cross-multiply the equation to get rid of the denominators:

\[
x^2(x - 2) = x^3(x - 1)
\]

Expand both sides:

\[
x^2 \cdot (x - 2) = x^3 \cdot (x - 1)
\]

\[
x^3 - 2x^2 = x^4 - x^3
\]

### Step 2: Move all terms to one side
To solve the equation, we move all terms to one side:

\[
x^3 - 2x^2 - x^4 + x^3 = 0
\]

Simplify:

\[
-x^4 + 2x^3 - 2x^2 = 0
\]

### Step 3: Factor the expression
Now, we factor out the common term, which is \( -x^2 \):

\[
-x^2(x^2 - 2x + 2) = 0
\]

This gives us two possible equations to solve:

1. \( -x^2 = 0 \)
2. \( x^2 - 2x + 2 = 0 \)

### Step 4: Solve each equation

#### For \( -x^2 = 0 \):
\[
x^2 = 0 \quad \Rightarrow \quad x = 0
\]

#### For \( x^2 - 2x + 2 = 0 \):
We solve this quadratic equation using the quadratic formula:

\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(2)}}{2(1)}
\]

\[
x = \frac{2 \pm \sqrt{4 - 8}}{2} = \frac{2 \pm \sqrt{-4}}{2}
\]

\[
x = \frac{2 \pm 2i}{2} = 1 \pm i
\]

Thus, the solutions are \( x = 1 + i \) and \( x = 1 - i \).

### Final Solutions:
The complete set of solutions is:
\[
x = 0, \, x = 1 + i, \, x = 1 - i
\]

Do you need more details or have any questions about the process?

Here are five related questions:
1. How do we know when to apply the quadratic formula?
2. What are imaginary numbers and why did they appear here?
3. How do we factor higher-order polynomials like \(x^4 - 2x^3 + 2x^2\)?
4. Can all quadratic equations have real solutions?
5. What happens if the discriminant of a quadratic equation is negative?

**Tip:** Always check for imaginary or complex solutions when dealing with quadratic equations with negative discriminants.

x²/(x-1)=x³/(x-2) x²(x-2)=x³(x-1) x³-2x²=x⁴-x³ -x⁴+2x³-2x²=0

This detailed guide walks through solving the polynomial equation x²/(x-1) = x³/(x-2), including cross-multiplication, factorization, and complex solutions using the quadratic formula. Learn key algebraic concepts and work through imaginary number solutions. Suitable for high school students in Grades 10-12.

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