Question 133465
The problem that you are having is that there is no real number x such that {{{x^2=-1}}}, hence you are doomed to eternal frustration trying to find a real number solution to your problem.  In order to deal with this problem, we created a number that is not a real number and called it an imaginary number represented by <i>i</i> and defined by the relationship: {{{i^2=-1}}}.


Using <i>i</i>, you can solve your problem.


{{{p^2 + 100 = 0}}}


{{{p^2=-100}}}


{{{p^2=(-1)(100)}}}


{{{p=sqrt(-1)*sqrt(100)}}}  or {{{p=-sqrt(-1)*sqrt(100)}}} (remember, you must consider both the positive and negative square root because if {{{x^2=4}}}, then {{{x=2}}} or {{{x=-2}}}.


Since {{{i^2=-1}}}, {{{i=sqrt(-1)}}}, so:


{{{p=i*sqrt(100)}}}  or {{{p=-i*sqrt(100)}}}
{{{p=10i}}}  or {{{p=-10i}}}


Let's check the answer:
{{{(10i)^2=100i^2}}}, but {{{i^2=-1}}}, so {{{100i^2=-100}}}  and


{{{(-10i)^2=100i^2}}}, but {{{i^2=-1}}}, so {{{100i^2=-100}}} 


Answer checks.


In general, anytime the value under a radical is less than zero, factor out a -1 and the square root of that is <i>i</i>, then take the positive square root of the positive number remaining under the radical.  The square root is then the square root of the positive value times <i>i</i>.