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 \(7z^2 - 20z + 14 = 0\).