Question

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

Answer 1

`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)`