Question

How do you solve `X^2-2x=-2`?

Answer 1

There are no Real solutions to this equation.
If Complex solutions are permitted: `x=1+-i`

Explanation:

Given
`color(white)("XXX")x^2-2x=-2`

Complete the square
#color(white)("XXX")x^2-2x+1 = -2+1

Re-write as a squared binomial and simplify the right side:
#color(white)("XXX")(x-1)^2 = -1

At this point we can see that there are no Real solutions (since any Real value squared is `> -1`)

If we allow Complex solutions:
`color(white)("XXX")x-1 = +-sqrt(-1) = +-i`

And adding `1` to both sides:
`color(white)("XXX")x= 1+-i`

Answer 2

There are no Real solutions to this equation.
If Complex solutions are permitted: `x=1+-i`

Explanation:

Given
`color(white)("XXX")x^2-2x=-2`

Complete the square
#color(white)("XXX")x^2-2x+1 = -2+1

Re-write as a squared binomial and simplify the right side:
#color(white)("XXX")(x-1)^2 = -1

At this point we can see that there are no Real solutions (since any Real value squared is `> -1`)

If we allow Complex solutions:
`color(white)("XXX")x-1 = +-sqrt(-1) = +-i`

And adding `1` to both sides:
`color(white)("XXX")x= 1+-i`