# How do you factor  d^2 + 2d + 2?

Oct 10, 2016

This quadratic only factorises if you use Complex coefficients:

${d}^{2} + 2 d + 2 = \left(d + 1 - i\right) \left(d + 1 + i\right)$

#### Explanation:

Completing the square, we find:

${d}^{2} + 2 d + 2 = {\left(d + 1\right)}^{2} + 1$

This will be positive and therefore non-zero for any Real value of $d$. So this expression is not reducible into linear factors with Real coefficients.

If we allow Complex numbers then this can be factored as a difference of squares.

${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

with $a = \left(d + 1\right)$ and $b = i$ as follows:

${d}^{2} + 2 d + 2 = {\left(d + 1\right)}^{2} + 1$

$\textcolor{w h i t e}{{d}^{2} + 2 d + 2} = {\left(d + 1\right)}^{2} - {i}^{2}$

$\textcolor{w h i t e}{{d}^{2} + 2 d + 2} = \left(\left(d + 1\right) - i\right) \left(\left(d + 1\right) + i\right)$

$\textcolor{w h i t e}{{d}^{2} + 2 d + 2} = \left(d + 1 - i\right) \left(d + 1 + i\right)$