4K^2-6k-5=0

1. Identify the coefficients:

a = 4, \hspace{5pt} b = -6, \hspace{5pt} c = -5

2. Use the quadratic formula:

k=\frac{-b \pm\sqrt{b^2 - 4ac}}{2a}

3. Calculate discriminant:

b^2 - 4ac = (-6)^2 - 4 \cdot 4 \cdot (-5) = 36 + 80 = 116

4. Simplify:

k=\frac{6 \pm\sqrt{116}}{8}=\frac{6 \pm2\sqrt{29}}{8}

5. Final answer:

k=\frac{3 \pm\sqrt{29}}{4}