To solve the equation \( x^2 - 10x + 29 = 0 \), we'll use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this equation, \( a = 1 \), \( b = -10 \), and \( c = 29 \).
First, we calculate the discriminant:
\[ b^2 - 4ac = (-10)^2 - 4 \cdot 1 \cdot 29 = 100 - 116 = -16 \]
Since the discriminant is negative, the solutions will involve imaginary numbers. Now we plug in the values into the quadratic formula:
\[ x = \frac{-(-10) \pm \sqrt{-16}}{2 \cdot 1} = \frac{10 \pm \sqrt{-16}}{2} \]
We know that \( \sqrt{-16} = 4i \), so:
\[ x = \frac{10 \pm 4i}{2} = 5 \pm 2i \]
Thus, the solutions to the equation are
\[ x = 5 + 2i \quad \text{and} \quad x = 5 - 2i \]
This matches with option A:
A. \( x = 5 \pm 2i \)