Solve for xx when f(x)=g(x)f(x)=g(x) if

f(x)=xx−6+3x−7f(x)=xx-6+3x-7 and g(x)=−10x−26x2−13x+42g(x)=-10x-26x2-13x+42


We are tasked with solving for \( x \) when \( f(x) = g(x) \), given the following functions:

\[
f(x) = \frac{x}{x - 6} + 3x - 7
\]
and
\[
g(x) = \frac{-10x - 26}{x^2 - 13x + 42}
\]

To find where \( f(x) = g(x) \), we need to set the two expressions equal to each other and solve for \( x \):

\[
\frac{x}{x - 6} + 3x - 7 = \frac{-10x - 26}{x^2 - 13x + 42}
\]

### Steps to solve:

1. **Rewrite the equation**:
   \[
   \frac{x}{x - 6} + 3x - 7 = \frac{-10x - 26}{x^2 - 13x + 42}
   \]
   
2. **Find a common denominator**:
   Multiply both sides by \( (x - 6)(x^2 - 13x + 42) \) to eliminate the denominators.

3. **Simplify the equation**:
   After eliminating the denominators, simplify the resulting equation.

4. **Solve for \( x \)**:
   Solve the resulting equation for \( x \), checking for any restrictions due to the original denominators.

Let me work through the algebra to solve this.The exact solutions for \( x \) involve complex expressions with cube roots and imaginary components. These solutions are:

\[
x_1 = 5 - \frac{5}{3(-\frac{1}{2} - \frac{\sqrt{3}i}{2})(2\sqrt{34009} + 369)^{\frac{1}{3}}} - \frac{(-\frac{1}{2} - \frac{\sqrt{3}i}{2})(2\sqrt{34009} + 369)^{\frac{1}{3}}}{3}
\]

\[
x_2 = 5 - \frac{(-\frac{1}{2} + \frac{\sqrt{3}i}{2})(2\sqrt{34009} + 369)^{\frac{1}{3}}}{3} - \frac{5}{3(-\frac{1}{2} + \frac{\sqrt{3}i}{2})(2\sqrt{34009} + 369)^{\frac{1}{3}}}
\]

\[
x_3 = -\frac{(2\sqrt{34009} + 369)^{\frac{1}{3}}}{3} - \frac{5}{3(2\sqrt{34009} + 369)^{\frac{1}{3}}} + 5
\]

These solutions are quite complex and involve imaginary components. To obtain an approximate numerical solution or simplify further, we would likely need numerical methods.

Would you like me to provide a numerical approximation for these solutions, or do you have specific questions about the result?

---

Here are five follow-up questions to deepen understanding:
1. What is the importance of domain restrictions in rational functions like \( f(x) \) and \( g(x) \)?
2. How does solving rational equations differ from solving polynomial equations?
3. Why do some equations involve complex solutions, and how do we interpret them?
4. How do singularities (denominators equating to zero) affect solutions in rational equations?
5. How can numerical methods simplify solving complex algebraic equations?

**Tip**: Always check the domain of rational functions to avoid division by zero, which can invalidate solutions.

Solve for x when f(x) = g(x) if f(x) = x / (x - 6) + 3x - 7 and g(x) = (-10x - 26) / (x^2 - 13x + 42)

This problem focuses on solving a rational equation where two functions are set equal to each other. The solution includes finding common denominators, simplifying complex expressions, and understanding when complex numbers arise as part of the solution. Ideal for advanced high school or college-level students.

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