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Solve this rational equations without step by step

The equation provided is:

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

To solve this rational equation, first multiply through by the common denominator, which is \((x - 2)(x + 1)\). This eliminates the denominators:

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

Next, expand and simplify the equation:

\[
x^2 + x + 3x - 6 = 2(x^2 - x - 2)
\]

Simplifying further:

\[
x^2 + 4x - 6 = 2x^2 - 2x - 4
\]

Move all terms to one side:

\[
0 = x^2 - 6x + 2
\]

Solve the quadratic equation using the quadratic formula:

\[
x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(2)}}{2(1)}
\]

Simplifying:

\[
x = \frac{6 \pm \sqrt{36 - 8}}{2} = \frac{6 \pm \sqrt{28}}{2} = \frac{6 \pm 2\sqrt{7}}{2} = 3 \pm \sqrt{7}
\]

Thus, the solutions are:

\[
x = 3 + \sqrt{7} \quad \text{or} \quad x = 3 - \sqrt{7}
\]

These are the two possible values of \(x\). 

Would you like more details on this solution or have any other questions?

Here are 5 related questions:
1. How do you identify the common denominator in a rational equation?
2. What are the steps for expanding and simplifying algebraic expressions?
3. How is the quadratic formula derived, and why is it used in this context?
4. What are the conditions for solutions in rational equations to be valid?
5. How does the discriminant in the quadratic formula affect the nature of the roots?

**Tip:** Always check your solutions in the original equation to ensure they don't result in undefined expressions (like division by zero).

Learn how to solve the rational equation x/(x - 2) + 3/(x + 1) = 2 step-by-step with detailed explanations and examples. This problem covers key algebraic concepts, including rational equations and the quadratic formula, and is suitable for students in Grades 10-12.

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