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

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!