7z^2-20z+14=0

To solve the quadratic equation \(7z^2 - 20z + 14 = 0\), we can use the quadratic formula: \[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \(a = 7\), \(b = -20\), and \(c = 14\). 1. Calculate the discriminant: \[ b^2 - 4ac = (-20)^2 - 4 \cdot 7 \cdot 14 \] \[ = 400 - 392 = 8 \] 2. Now, substitute the values into the quadratic formula: \[ z = \frac{-(-20) \pm \sqrt{8}}{2 \cdot 7} \] \[ = \frac{20 \pm \sqrt{8}}{14} \] Since \(\sqrt{8} = 2\sqrt{2}\), we can rewrite it as: \[ z = \frac{20 \pm 2\sqrt{2}}{14} \] 3. Simplifying this expression gives: \[ z = \frac{10 \pm \sqrt{2}}{7} \] Thus, the two solutions are: \[ z_1 = \frac{10 + \sqrt{2}}{7} \] \[ z_2 = \frac{10 - \sqrt{2}}{7} \] These are the solutions to the equation \(7z^2 - 20z + 14 = 0\).