To find the roots of the quadratic equation [tex]\(y = x^2 - 10x + 125\)[/tex], we start by identifying the coefficients:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -10\)[/tex]
- [tex]\(c = 125\)[/tex]
The roots of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] are given by the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
First, calculate the discriminant [tex]\(D\)[/tex], which is [tex]\(b^2 - 4ac\)[/tex]:
[tex]\[
D = (-10)^2 - 4 \cdot 1 \cdot 125 = 100 - 500 = -400
\][/tex]
Since the discriminant is negative, the quadratic equation has complex roots. Complex roots arise when the square root of the discriminant includes an imaginary number.
We start with the real part of the roots:
[tex]\[
\text{Real part} = \frac{-b}{2a} = \frac{-(-10)}{2 \cdot 1} = \frac{10}{2} = 5
\][/tex]
Next, we calculate the imaginary part using the absolute value of the discriminant:
[tex]\[
\text{Imaginary part} = \frac{\sqrt{|D|}}{2a} = \frac{\sqrt{400}}{2 \cdot 1} = \frac{20}{2} = 10
\][/tex]
Thus, the roots of the equation are:
[tex]\[
x = \text{Real part} \pm i \cdot \text{Imaginary part} = 5 \pm 10i
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{x = 5 \pm 10i}
\][/tex]
So, the correct option is B.
