How do you solve -4n ^ { 2} + 2n - 5= 0?

1 Answer
George C.
Aug 9, 2018

n = 1/4+-sqrt(19)/4i

Explanation:

We can solve this by completing the square and using the imaginary unit i, which satisfies i^2=-1. First multiply by -4 so we can do much of the arithmetic using integers...

0 = -4(-4n^2+2n-5)

color(white)(0) = 16n^2-8n+1+19

color(white)(0) = (4n-1)^2+(sqrt(19))^2

color(white)(0) = (4n-1)^2-(sqrt(19)i)^2

color(white)(0) = ((4n-1)-sqrt(19)i)((4n-1)+sqrt(19)i)

color(white)(0) = (4n-1-sqrt(19)i)(4n-1+sqrt(19)i)

Hence:

4n = 1+-sqrt(19)i

So:

n = 1/4+-sqrt(19)/4i

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