https://file.aiquickdraw.com/mathbot/file/1722961618754ye38mxue.jpg

The problem in the image is:

\[ \frac{x^2}{x-1} = \frac{x+2}{2x-5} \]

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

1. **Cross-multiply the fractions** to eliminate the denominators:
\[ (x^2)(2x-5) = (x-1)(x+2) \]

2. **Expand both sides of the equation**:
\[ 2x^3 - 5x^2 = x^2 + 2x - 1 \]

3. **Combine like terms** to form a polynomial equation:
\[ 2x^3 - 5x^2 - x^2 - 2x + 1 = 0 \]
\[ 2x^3 - 6x^2 - 2x + 1 = 0 \]

4. **Divide the entire equation by the common factor** (if any) to simplify:
The equation is already simplified.

5. **Factor the polynomial equation** or use numerical methods to find the roots:
\[ 2x^3 - 6x^2 - 2x + 1 = 0 \]

Finding the roots of a cubic equation can be complex, so we can use numerical methods or a graphing calculator to find approximate solutions.

However, to proceed with factoring if possible, we might need to use the Rational Root Theorem to test for potential rational roots or use synthetic division.

Alternatively, we can use the **numerical approximation**:

**Numerical approximation:**

Using a numerical solver or graphing, we find the approximate roots of the equation \( 2x^3 - 6x^2 - 2x + 1 = 0 \).

Let's use Python to find the roots of the polynomial:

```python
import numpy as np

# Coefficients of the polynomial 2x^3 - 6x^2 - 2x + 1
coefficients = [2, -6, -2, 1]

# Finding the roots of the polynomial
roots = np.roots(coefficients)
roots
```

Let's compute this.The roots of the polynomial \( 2x^3 - 6x^2 - 2x + 1 = 0 \) are approximately:

1. \( x \approx 3.260 \)
2. \( x \approx -0.542 \)
3. \( x \approx 0.283 \)

These are the values of \( x \) that satisfy the given equation.

Do you have any questions or would you like more details about any part of the solution?

Here are 8 related questions to expand on this topic:

1. How do you use the Rational Root Theorem to find potential rational roots?
2. What are the steps to solve a quadratic equation?
3. How do you factor a cubic polynomial?
4. What methods can be used to solve higher-degree polynomials?
5. How does synthetic division help in finding the roots of a polynomial?
6. What are the applications of finding polynomial roots in real-world problems?
7. How can graphing polynomials help in understanding their behavior?
8. What is the importance of the Fundamental Theorem of Algebra?

**Tip:** When solving polynomial equations, always check for potential rational roots using the Rational Root Theorem before resorting to numerical methods. This can often simplify the problem significantly.

Learn how to solve the cubic equation 2x^3 - 6x^2 - 2x + 1 step-by-step. This problem covers equations, rational roots, polynomials, and numerical methods, including a detailed solution and approximate roots. Suitable for advanced high school students in Grades 11-12.

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