How do you factor the trinomial #a^2 - a + 43#?

1 Answer
George C.
May 6, 2016

#a^2-a+43=(a-1/2-(3sqrt(19))/2i)(a-1/2+(3sqrt(19))/2i)#

Explanation:

Complete the square and use the difference of squares identity:

#A^2-B^2=(A-B)(A+B)#

with #A=(a-1/2)# and #B=(3sqrt(19))/2 i# as follows:

#a^2-a+43#

#=(a-1/2)^2-1/4+43#

#=(a-1/2)^2-1/4+172/4#

#=(a-1/2)^2+171/4#

#=(a-1/2)^2+(9*19)/4#

#=(a-1/2)^2-((3sqrt(19))/2i)^2#

#=((a-1/2)-(3sqrt(19))/2i)((a-1/2)+(3sqrt(19))/2i)#

#=(a-1/2-(3sqrt(19))/2i)(a-1/2+(3sqrt(19))/2i)#

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