Question

How do you solve the quadratic `u^2-2u+3=0` using any method?

Answer 1

`u = 1+-sqrt(2)i`

Explanation:

The difference of squares identity can be written:

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

Use this with `a=(u-1)` and `b=sqrt(2)i` as follows:

`0 = u^2-2u+3`

`color(white)(0) = u^2-2u+1+2`

`color(white)(0) = (u-1)^2-(sqrt(2)i)^2`

`color(white)(0) = ((u-1)-sqrt(2)i)((u-1)+sqrt(2)i)`

`color(white)(0) = (u-1-sqrt(2)i)(u-1+sqrt(2)i)`

Hence `u = 1+-sqrt(2)i`