To solve the quadratic equation \( x^2 - 14x + 33 = 0 \), we can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a = 1 \), \( b = -14 \), and \( c = 33 \).

First, calculate the discriminant:

\[
b^2 - 4ac = (-14)^2 - 4 \cdot 1 \cdot 33 = 196 - 132 = 64
\]

Since the discriminant is positive, there are two distinct real roots. Now we plug the values into the quadratic formula:

\[
x = \frac{-(-14) \pm \sqrt{64}}{2 \cdot 1}
\]

\[
x = \frac{14 \pm 8}{2}
\]

This gives us the two solutions:

\[
x_1 = \frac{14 + 8}{2} = \frac{22}{2} = 11
\]

\[
x_2 = \frac{14 - 8}{2} = \frac{6}{2} = 3
\]

Thus, the solutions to the equation \( x^2 - 14x + 33 = 0 \) are \( x = 11 \) and \( x = 3 \).