To solve the quadratic equation \(x^2 - 20x + 34 = 0\), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 1\), \(b = -20\), and \(c = 34\).
First, calculate the discriminant (\(b^2 - 4ac\)):
\[ b^2 = (-20)^2 = 400 \] \[ 4ac = 4 \times 1 \times 34 = 136 \] \[ b^2 - 4ac = 400 - 136 = 264 \]
Now substitute the values into the quadratic formula:
\[ x = \frac{-(-20) \pm \sqrt{264}}{2 \cdot 1} \] \[ x = \frac{20 \pm \sqrt{264}}{2} \]
Next, simplify \(\sqrt{264}\):
\[ \sqrt{264} = \sqrt{4 \times 66} = 2\sqrt{66} \]
Substituting this back into the equation gives:
\[ x = \frac{20 \pm 2\sqrt{66}}{2} \]
Simplifying further:
\[ x = 10 \pm \sqrt{66} \]
Thus, the solutions to the equation \(x^2 - 20x + 34 = 0\) are:
\[ x = 10 + \sqrt{66} \quad \text{and} \quad x = 10 - \sqrt{66} \]