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You've requested help with problem number 2 from your worksheet, which involves solving the equation:

\[
\frac{x^2 + 10x + 39}{x^2 + 4x + 2} = 2
\]

To solve this equation, we'll first isolate terms to get a single quadratic equation. Here are the steps:

1. **Set Up the Equation**:
   \[
   \frac{x^2 + 10x + 39}{x^2 + 4x + 2} = 2
   \]

2. **Eliminate the Denominator**:
   \[
   x^2 + 10x + 39 = 2(x^2 + 4x + 2)
   \]

3. **Expand the Right Side**:
   \[
   x^2 + 10x + 39 = 2x^2 + 8x + 4
   \]

4. **Rearrange Terms to Form a Quadratic Equation**:
   \[
   0 = x^2 - 2x - 35
   \]

5. **Factorize the Quadratic Equation**:
   \[
   0 = (x - 7)(x + 5)
   \]

6. **Solve for x**:
   \[
   x = 7 \quad \text{or} \quad x = -5
   \]

Therefore, the solutions to the equation are \( x = 7 \) and \( x = -5 \).

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

**Here are some related questions to explore:**
1. What happens if you substitute \( x = 7 \) or \( x = -5 \) back into the original equation?
2. How do you determine if the original rational function has any vertical asymptotes?
3. Can you find the horizontal asymptote of the rational function?
4. What is the domain of the original rational function?
5. How would you graph the rational function to verify the solution?

**Tip:** Always verify your solutions by substituting them back into the original equation to ensure they satisfy the equation, especially for rational equations where undefined points may be critical.

Travail 2: Soit f la fonction définie sur ℝ par f(x) = 1/2 * (x^2 + 4x - 10).

Learn how to solve and analyze the quadratic function f(x) = 1/2(x^2 + 4x - 10). This problem involves working with quadratic equations, solving inequalities, and finding the minimum value. Suitable for high school students in Grades 10-12.

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