Question 736993
{{{ 3y^2 - 4y - 1 = 0 }}}
{{{ 3y^2 - 4y = 1 }}}
{{{ y^2 - (4/3)*y = 1/3 }}}
In this equation,  which is for finding the roots,
{{{ y }}} is called the independent variable.
-------------------
It would be plotted on the horizontal axis, 
which is normally the {{{ x }}} axis.
-------------------
The rule for completing the square is:
Take 1/2 of the co-efficient of the {{{ y }}}
term ( usually called the {{{ x }}} term  ),
square it, then add it to both sides
-------------------
The co-efficient of the {{{ y }}} term is
{{{ -4/3 }}}, so
{{{ (-4/3) / 2 = -4/6 }}}
{{{ -4/6 = -2/3 }}}
Now square it
{{{ (-2/3 )^2 = 4/9 }}}
Add it to both sides
{{{ y^2 - (4/3)*y  + 4/9 = 1/3 + 4/9 }}}
{{{ y^2 - (4/3)*y  + 4/9 = 3/9 + 4/9 }}}
{{{ y^2 - (4/3)*y  + 4/9 = 7/9 }}}
{{{ ( y - 2/3 )^2 = ( sqrt(7) / 3 )^2 }}}
Take the square root of both sides
{{{ y - 2/3 = sqrt(7) / 3 }}}
{{{ y = ( 2 + sqrt(7) ) / 3 }}}
and
{{{ y = ( 2 - sqrt(7) ) / 3 }}}
------------------------
Here's a plot of the entire equation, which can
be expressed as {{{ z = 3y^2 - 4y - 1 }}}
Note that {{{ sqrt(7) = 2.6458 }}}
{{{ ( 2 + 2.6458 ) / 3 = 1.5486 }}} and
{{{ ( 2 - 2.6458 ) / 3 = -.2153 }}}
You can see that these are the horizontal axis crossings
( Unless I messed up )
{{{ graph( 400, 400, -4, 4, -4, 6, 3x^2 - 4x - 1 ) }}}