To solve the quadratic equation [tex]\(x^2 - 14x + 58 = 0\)[/tex], we'll use the quadratic formula, which states that for any quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex], the solutions for [tex]\(x\)[/tex] are given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For our specific equation [tex]\(x^2 - 14x + 58 = 0\)[/tex]:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -14\)[/tex]
- [tex]\(c = 58\)[/tex]
First, calculate the discriminant, which is [tex]\(b^2 - 4ac\)[/tex]:
[tex]\[
b^2 - 4ac = (-14)^2 - 4 \cdot 1 \cdot 58 = 196 - 232 = -36
\][/tex]
Since the discriminant is negative, the solutions will be complex numbers.
Next, apply the quadratic formula:
[tex]\[
x = \frac{-(-14) \pm \sqrt{-36}}{2 \cdot 1} = \frac{14 \pm \sqrt{-36}}{2} = \frac{14 \pm 6i}{2}
\][/tex]
This simplifies to:
[tex]\[
x = \frac{14}{2} \pm \frac{6i}{2} = 7 \pm 3i
\][/tex]
Thus, the solutions are:
[tex]\[
x = 7 + 3i \quad \text{and} \quad x = 7 - 3i
\][/tex]
Therefore, the correct answer is:
D. [tex]\(\{7+3i, 7-3i\}\)[/tex]
