Question 571216
{{{ 3x^2 - x - 1 = 0 }}}
{{{ 3x^2 - x = 1 }}}
{{{ x^2 -(1/3)*x = 1/3 }}}
Take 1/2 of the coefficient of {{{x}}}, square it
and add it to both sides
{{{ x^2 -(1/3)*x +  (-1/6)^2 = 1/3  + (-1/6)^2 }}}
{{{ x^2 -(1/3)*x + 1/36 = 12/36 + 1/36 }}}
{{{ ( x - 1/6 )^2 = 13/36 }}}
Take the square root of both sides
{{{ x - 1/6 = sqrt( 13 ) / 6 }}}
{{{ x = ( 1 + sqrt(13) ) / 6 }}}
and, also
{{{ x = ( 1 - sqrt(13) ) / 6 }}}
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check the solutions:
{{{ 3*( ( 1 - sqrt(13) ) / 6 ) )^2 - ( 1 - sqrt(13) ) / 6 ) - 1 = 0 }}}
{{{ (3/36)*( 1 - 2*sqrt(13) + 13 )  - 1/6 + ( sqrt(13))/6  - 1 = 0 }}}
Multiply both sides by {{{ 12 }}}
{{{ 1 - 2*sqrt(13) + 13 - 2 + 2*sqrt(13) - 12 = 0 }}}
{{{ 1 + 13 - 2 - 12 = 0 }}}
{{{ 0 = 0 }}}
OK. You can check other solution