Solve the equation by the quadratic formula. \( 15 x^{2}-17 x-4=0 \)

To solve the quadratic equation \(15 x^{2}-17 x-4=0\) using the quadratic formula, we first identify the coefficients: \(a = 15\), \(b = -17\), and \(c = -4\). The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Plugging in the coefficients: 1. Calculate \(b^2 - 4ac\): \[ b^2 = (-17)^2 = 289 \] \[ 4ac = 4 \cdot 15 \cdot (-4) = -240 \] Therefore, \[ b^2 - 4ac = 289 - (-240) = 289 + 240 = 529 \] 2. Now, apply the quadratic formula: \[ x = \frac{17 \pm \sqrt{529}}{2 \cdot 15} \] Since \( \sqrt{529} = 23\): \[ x = \frac{17 \pm 23}{30} \] 3. This gives us two solutions: \[ x_1 = \frac{17 + 23}{30} = \frac{40}{30} = \frac{4}{3} \] \[ x_2 = \frac{17 - 23}{30} = \frac{-6}{30} = -\frac{1}{5} \] Thus, the solutions are \(x = \frac{4}{3}\) and \(x = -\frac{1}{5}\).