$$ x = \frac { - 3 \pm \sqrt { 13 - 2 k } } { 4 } $$

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The expression you've provided is a solution for $$ x $$ in terms of $$ k $$. It appears to be derived from a quadratic equation. Let's analyze it step by step.

1. **Identify the components**:
   - The expression is in the form of a quadratic formula: 
     $$
     x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
     $$
   - Here, $$ a = 4 $$, $$ b = 3 $$, and the discriminant $$ b^2 - 4ac = 13 - 2k $$.

2. **Understanding the discriminant**:
   - The term $$ 13 - 2k $$ must be non-negative for $$ x $$ to have real solutions. This gives us the condition:
     $$
     13 - 2k \geq 0
     $$
   - Solving this inequality:
     $$
     2k \leq 13 \implies k \leq \frac{13}{2} = 6.5
     $$

3. **Final expression**:
   - The expression for $$ x $$ can take two values depending on the sign in front of the square root:
     $$
     x_1 = \frac{-3 + \sqrt{13 - 2k}}{4}, \quad x_2 = \frac{-3 - \sqrt{13 - 2k}}{4}
     $$

4. **Conclusion**:
   - The values of $$ x $$ depend on the value of $$ k $$ and are valid as long as $$ k \leq 6.5 $$.

If you have a specific question or need further calculations regarding this expression, please let me know!
The solutions for $$ x $$ are: $$ x = \frac{-3 \pm \sqrt{13 - 2k}}{4} $$ These solutions are valid when $$ k \leq 6.5 $$.

\( x = \frac { - 3 \pm \sqrt { 13 - 2 k } } { 4 } \)

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