Question 632148
Let {{{ w^4 = z^2 }}}
{{{ z = w^2 }}} and
{{{ z = -w^2 }}}

{{{ w^4 - 20w^2 - 2 = 0  }}}
{{{ z^2 - 20z  - 2 = 0 }}}
{{{ z^2 - 20z = 2 }}}
Complete the square
{{{ z^2 - 20z + (-20/2)^2 = 2 + (-20/2)^2 }}}
{{{ z^2 - 20z + 100  = 2 + 100 }}}
{{{ ( z - 10 )^2 = 102 }}}
Take the square root of both sides
{{{ z - 10 = sqrt(102) }}}
{{{ z = 10 + sqrt( 102 ) }}}
and
{{{ z = 10 - sqrt( 102 ) }}}
----------------------
{{{ w^2 = 10 + sqrt( 102 ) }}}
{{{ w = sqrt( 10 + sqrt(102) ) }}}
{{{ w = -sqrt( 10 + sqrt(102) ) }}}
-----------------------
{{{ w^2 = 10 - sqrt(102) }}}
{{{ w = sqrt( 10 - sqrt(102) ) }}}
{{{ w = -sqrt( 10 - sqrt(102) ) }}}
-----------------------
These are the 4 roots
Here's a plot of the equation:
{{{ graph( 400, 400, -8, 8, -120, 20, x^4 - 20x^2 - 2 ) }}}
There are 2 real roots and 2 imaginary, which
seems to agree with the plot