https://file.aiquickdraw.com/mathbot/file/17246819458758zdivk93.jpg

The equation in the image is:

\[
\frac{(x - 2)(x - 3)}{(x - 1)(x + 2)} = \frac{2}{3}
\]

This equation is already factored, but let's solve it to see if any further simplification or factoring is needed.

### Steps to Solve:

1. **Cross Multiply**: 
   \[
   3(x - 2)(x - 3) = 2(x - 1)(x + 2)
   \]

2. **Expand Both Sides**:
   - Left side: 
     \[
     3[(x - 2)(x - 3)] = 3[x^2 - 5x + 6] = 3x^2 - 15x + 18
     \]
   - Right side: 
     \[
     2[(x - 1)(x + 2)] = 2[x^2 + x - 2] = 2x^2 + 2x - 4
     \]

3. **Set the Equation to Zero**:
   \[
   3x^2 - 15x + 18 - 2x^2 - 2x + 4 = 0
   \]
   Simplifying this:
   \[
   x^2 - 17x + 22 = 0
   \]

4. **Factor the Quadratic Equation**:
   We now need to factor \(x^2 - 17x + 22 = 0\).

   This quadratic doesn't factor nicely with integers, so we use the quadratic formula:

   \[
   x = \frac{-(-17) \pm \sqrt{(-17)^2 - 4(1)(22)}}{2(1)}
   \]
   Simplifying:
   \[
   x = \frac{17 \pm \sqrt{289 - 88}}{2}
   \]
   \[
   x = \frac{17 \pm \sqrt{201}}{2}
   \]
   \[
   x = \frac{17 \pm \sqrt{201}}{2}
   \]

So the roots are \( x = \frac{17 + \sqrt{201}}{2} \) and \( x = \frac{17 - \sqrt{201}}{2} \). These are the values for \( x \).

Would you like any further details, or have any other questions on this problem?

Here are 5 related questions to consider:
1. How do you factorize quadratic expressions in general?
2. What is the significance of the discriminant in a quadratic equation?
3. How does cross-multiplication work when solving rational equations?
4. What are the conditions for a quadratic equation to have real roots?
5. How would you solve the equation if there were more complex factors in the numerator or denominator?

**Tip**: Always check if a quadratic equation can be factored easily with integers before using the quadratic formula. It saves time!

Learn how to solve the rational equation (x - 2)(x - 3) / (x - 1)(x + 2) = 2/3 with a step-by-step solution. This problem covers key algebraic concepts including factoring, cross-multiplication, and solving rational equations. Suitable for students in Grades 10-12.

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