To solve the equation \(x^2 = 19x + 1\), we first rearrange it into standard form:
\[ x^2 - 19x - 1 = 0. \]
Next, we apply the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \]
where \(a = 1\), \(b = -19\), and \(c = -1\). Plugging in these values gives:
\[ x = \frac{-(-19) \pm \sqrt{(-19)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1}. \]
Now, calculating the components:
- Calculate \(b^2\):
\[ (-19)^2 = 361. \]
- Calculate \(4ac\):
\[ 4 \cdot 1 \cdot (-1) = -4. \]
- Now compute \(b^2 - 4ac\):
\[ 361 - (-4) = 361 + 4 = 365. \]
Now, substituting back into the formula:
\[ x = \frac{19 \pm \sqrt{365}}{2}. \]
Thus, the solutions to the equation are:
\[ x = \frac{19 - \sqrt{365}}{2} \quad \text{and} \quad x = \frac{19 + \sqrt{365}}{2}. \]
Given the options, the appropriate pair of solutions is:
\[ \frac{19 - \sqrt{365}}{2}, \frac{19 + \sqrt{365}}{2}. \]
So the correct answer is:
\[ \text{StartFraction } 19 - \text{StartRoot } 365 \text{ EndRoot} \text{ Over } 2 \text{ EndFraction, \text{ StartFraction } 19 + \text{StartRoot } 365 \text{ EndRoot} \text{ Over } 2 \text{ EndFraction}. \]