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

Oct 1, 2015

There are no Real solutions to this equation.
If Complex solutions are permitted: $x = 1 \pm i$

#### Explanation:

Given
$\textcolor{w h i t e}{\text{XXX}} {x}^{2} - 2 x = - 2$

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:
$\textcolor{w h i t e}{\text{XXX}} x - 1 = \pm \sqrt{- 1} = \pm i$

And adding $1$ to both sides:
$\textcolor{w h i t e}{\text{XXX}} x = 1 \pm i$

Oct 1, 2015

There are no Real solutions to this equation.
If Complex solutions are permitted: $x = 1 \pm i$

#### Explanation:

Given
$\textcolor{w h i t e}{\text{XXX}} {x}^{2} - 2 x = - 2$

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:
$\textcolor{w h i t e}{\text{XXX}} x - 1 = \pm \sqrt{- 1} = \pm i$

And adding $1$ to both sides:
$\textcolor{w h i t e}{\text{XXX}} x = 1 \pm i$