How do you factor completely: $ {x^2} - 2x + 5 $ ?

Hint:In this question covert given expression into a quadratic equation and then factor it completely. We can find the factor of this quadratic equation by the factor method of finding roots for a quadratic equation. But since this equation does not have real roots we will do this by using quadratic formula.

Complete step by step solution:
Let us try to solve this question in which we need the factor of a given quadratic equation $ {x^2} - 2x + 5 = 0 $ .
To find the factor of this quadratic, we will first find the discriminant of this quadratic equation and from which we find the nature of the root of this quadratic equation. After which we will find the root of the quadratic equation using quadratic formula and factor it.

Types of root of quadratic equation: $ a{x^2} + bx + c = 0, $
1) Two distinct real roots, if $ {b^2} - 4ac > 0 $ ( which is called discriminant of this quadratic
equation)
2) Two equal real roots, if $ {b^2} - 4ac = 0 $
3) No real roots if, $ {b^2} - 4ac < 0 $
In this given quadratic equation $ {x^2} - 2x + 5 = 0 $
 $
a = 1 \\
b = - 2 \\
c = 5 \\
 $
So the discriminant of this quadratic equation is
 $ {b^2} - 4ac = {( - 2)^2} - 4 \cdot 1 \cdot 5 $
 $ = 4 - 20 = - 16 < 0 $

Since the value discriminant of this equation is less than 0, this quadratic equation has no real roots.

Now we will find the root of this quadratic equation by applying a quadratic formula.

Quadratic formula $ = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $

Putting value of $ a,b $ and $ c $ in the quadratic formula, we get
 \[
= \dfrac{{ - ( - 2) \pm \sqrt {{{( - 2)}^2} - 4 \cdot 1 \cdot 5} }}{{2 \cdot 1}} \\
= \dfrac{{2 \pm \sqrt {4 - 20} }}{2} \\
= \dfrac{{2 \pm \sqrt { - 16} }}{2} \\
\]

So as we know, $ \sqrt { - 1} = \iota $ we can write above equation by putting $ \sqrt { - 1} = \iota $ as
 $
= \dfrac{{2 \pm 4\iota }}{2} \\
= 1 \pm 2\iota \\

 $

So the root of the given quadratic equation $ {x^2} - 2x + 5 = 0 $ is $ x = 1 + 2\iota $ and $ x = 1 - 2\iota $ .

Hence the factor of this given expression is $ (x - 1 - 2\iota )(x + 1 - 2\iota ) $ . $ $

Note: While this we will first change the given expression into a quadratic equation and then check the nature of the root and then find the root of that quadratic equation using quadratic formula. To solve this type of question you need to know the conditions for the nature of roots.