# How do I solve (x-2)^2 = -3 using the square root property?

Aug 21, 2015

$x = 2 \pm \left(\sqrt{3}\right) i$

#### Explanation:

If ${a}^{2} = b$
then $a = \pm \sqrt{b}$

In this case $a = \left(x - 2\right)$ and $b = - 3$
So
$\textcolor{w h i t e}{\text{XXXX}} x - 2 = \pm \sqrt{- 3}$

Note that $\sqrt{- 3} = \sqrt{3} \cdot \sqrt{- 1}$
and there is no Real value $= \sqrt{- 1}$.
However, if we are allowed to use Complex values
by definition $i = \sqrt{- 1}$

Therefore, we can write
$\textcolor{w h i t e}{\text{XXXX}} x - 2 = \pm \left(\sqrt{3}\right) i$
and after adding 2 to both sides
$\textcolor{w h i t e}{\text{XXXX}} x = 2 \pm \left(\sqrt{3}\right) i$