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

1 Answer
George C.
Oct 10, 2016

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)#

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