What is the quadratic equation that has 2 – i as a root?

In a quadratic equation that has complex roots, the roots
are conjugates of each other. So, if one of the roots of the quadratic equation is x +
iy, the other root is x – yi.

Here we have one of the roots
as 2 – i, the other root is 2 + i

The quadratic equation
can be written as [x – (2 + i)] [x – (2 – i)] = 0

=>
[x – 2 – i] [x – 2 + i] = 0

=> x^2 – 2x + xi – 2x +
4 – 2i - xi + 2i – i^2 = 0

=> x^2 – 4x + 4 + 1 =
0

=> x^2 – 4x + 5 =
0

Therefore the quadratic equation is

x^2 – 4x + 5 =
0