How do you factor completely: x^2 − 2x + 5?

Jul 15, 2015

${x}^{2} - 2 x + 5 = \left(x - 1 + 2 i\right) \left(x - 1 - 2 i\right)$
$\textcolor{w h i t e}{\text{XXXX}}$(there are no Real factors for the given expression)

Explanation:

Write the expression as an equation with the expression equal to zero.
$\textcolor{w h i t e}{\text{XXXX}}$${x}^{2} - 2 x + 5 = 0$
Us the quadratic formula to determine the roots:
$\textcolor{w h i t e}{\text{XXXX}}$$\frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$
becomes
$\textcolor{w h i t e}{\text{XXXX}}$$1 \pm 2 i$

That is
$\textcolor{w h i t e}{\text{XXXX}}$$\left(x - \left(1 + 2 i\right)\right) \cdot \left(x - \left(1 - 2 i\right)\right) = 0$

So
$\textcolor{w h i t e}{\text{XXXX}}$$\left(x - 1 - 2 i\right) \mathmr{and} \left(x - 1 + 2 i\right)$ are factors of the the given expression.