Question

How do you solve `-5x^2-2x-4 = 0` ?

Answer 1

`x = 1/5(-1+-sqrt(19)i)`

Explanation:

Complete the square, then use the difference of squares identity:

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

with `a=(5x+1)` and `b=sqrt(19)i` as follows:

`0 = -5(-5x^2-2x-4)`

`color(white)(0) = 25x^2+10x+20`

`color(white)(0) = (5x)^2+2(5x)+1+19`

`color(white)(0) = (5x+1)^2+(sqrt(19))^2`

`color(white)(0) = (5x+1)^2-(sqrt(19)i)^2`

`color(white)(0) = ((5x+1)-sqrt(19)i)((5x+1)+sqrt(19)i)`

`color(white)(0) = (5x+1-sqrt(19)i)(5x+1+sqrt(19)i)`

Hence:

`5x = -1+-sqrt(19)i`

So:

`x = 1/5(-1+-sqrt(19)i)`