6. Solve \( \frac{b}{4}-\frac{5}{b}=\frac{1}{2} \) using the quadratic formula. Give the exact solution and approximate the solutions to three decimal places.

To solve the equation \( \frac{b}{4}-\frac{5}{b}=\frac{1}{2} \), first, clear the fractions by multiplying through by \( 4b \): \[ b^2 - 20 = 2b. \] Rearranging gives: \[ b^2 - 2b - 20 = 0. \] Now, we will use the quadratic formula \( b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \), where \( A = 1, B = -2, \) and \( C = -20 \): Calculating the discriminant: \[ B^2 - 4AC = (-2)^2 - 4(1)(-20) = 4 + 80 = 84. \] Now applying the quadratic formula: \[ b = \frac{-(-2) \pm \sqrt{84}}{2 \cdot 1} = \frac{2 \pm \sqrt{84}}{2} = 1 \pm \sqrt{21}. \] Thus, we have two exact solutions: \[ b = 1 + \sqrt{21} \quad \text{and} \quad b = 1 - \sqrt{21}. \] Now approximating \( \sqrt{21} \approx 4.583 \), we find: \[ b \approx 1 + 4.583 \approx 5.583 \quad \text{and} \quad b \approx 1 - 4.583 \approx -3.583. \] So the solutions are: \[ b \approx 5.583 \quad \text{and} \quad b \approx -3.583. \]