Question

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

Answer 1

This quadratic only factorises if you use Complex coefficients:

`d^2+2d+2 = (d+1-i)(d+1+i)`

Explanation:

Completing the square, we find:

`d^2+2d+2 = (d+1)^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 = (a-b)(a+b)`

with `a=(d+1)` and `b=i` as follows:

`d^2+2d+2 = (d+1)^2+1`

`color(white)(d^2+2d+2) = (d+1)^2-i^2`

`color(white)(d^2+2d+2) = ((d+1)-i)((d+1)+i)`

`color(white)(d^2+2d+2) = (d+1-i)(d+1+i)`