Question 10 Solve the equation \( 15 z^{2}-4 z-3=0 \). Answer: \( z= \) Write your answers as a list of integers or reduced fractions, with your ans For example, if you get 4 and \( -\frac{2}{3} \) as your answers, then enter \( 4,-2 / 3 \) in th

1. The quadratic equation is given by \[ 15 z^2 - 4z - 3 = 0. \] 2. Identify the coefficients: \[ a = 15, \quad b = -4, \quad c = -3. \] 3. Calculate the discriminant \(\Delta\): \[ \Delta = b^2 - 4ac = (-4)^2 - 4(15)(-3) = 16 + 180 = 196. \] 4. Compute the square root of the discriminant: \[ \sqrt{\Delta} = \sqrt{196} = 14. \] 5. Use the quadratic formula: \[ z = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-4) \pm 14}{2 \times 15} = \frac{4 \pm 14}{30}. \] 6. Solve for \(z\): - For the plus sign: \[ z = \frac{4 + 14}{30} = \frac{18}{30} = \frac{3}{5}. \] - For the minus sign: \[ z = \frac{4 - 14}{30} = \frac{-10}{30} = -\frac{1}{3}. \] 7. Final answers: \[ z = \frac{3}{5}, \quad z = -\frac{1}{3}. \]