Question

How do you simplify `x^2+x+2=0` ?

Answer 1

This will only "simplify" into linear factors with the help of Complex coefficients as:

`(x+1/2-sqrt(7)/2i)(x+1/2+sqrt(7)/2i) = 0`

Explanation:

Given:

`x^2+x+2 = 0`

Note that this is in the form:

`ax^2+bx+c = 0`

with `a=1`, `b=1` and `c=2`

This has discriminant `Delta` given by the formula:

`Delta = b^2-4ac = color(blue)(1)^2-4(color(blue)(1))(color(blue)(2)) = 1-8 = -7`

Since `Delta < 0` this quadratic has no simpler factors with Real coefficients.

We can simplify it with Complex coefficients by completing the square and using the difference of squares identity:

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

with `A=(x+1/2)` and `B=sqrt(7)/2i` as follows:

`0 = x^2+x+2`

`color(white)(0) = x^2+2(x)(1/2)+(1/2)^2+7/4`

`color(white)(0) = (x+1/2)^2-(sqrt(7)/2i)^2`

`color(white)(0) = ((x+1/2)-sqrt(7)/2i)((x+1/2)+sqrt(7)/2i)`

`color(white)(0) = (x+1/2-sqrt(7)/2i)(x+1/2+sqrt(7)/2i)`

Hence:

`x = -1/2+-sqrt(7)/2i`