Question 446754
if I say the roots are {{{ r[1] }}} and {{{ r[2] }}},
{{{ x = r[1] }}}
{{{ x = r[2] }}}
then
{{{ x - r[1] = 0 }}}
{{{ x - r[2] = 0 }}}
and
{{{ (x - r[1])*(x - r[2]) = 0 }}}
(1) {{{ x^2 - (r[1] + r[2])x + r[1]*r[2] = 0 }}}
given:
{{{ 8x² - 2x = 1 }}}
{{{ 8x^2 - 2x - 1 = 0 }}}
(2){{{ x^2 - (1/4)*x - 1/8 = 0 }}}
Comparing (2) with (1),
{{{ r[1] + r[2] = 1/4 }}}
The sum of the roots is 1/4
check:
Complete the square:
{{{ x^2 - (1/4)*x = 1/8  }}}
{{{ x^2 - (1/4)*x + (1/8)^2 = 1/8 + (1/8)^2 }}}
{{{ ( x - 1/8 )^2 = 8/64 + 1/64 }}}
{{{ ( x - 1/8 )^2 = 9/64 }}}
{{{ x - 1/8 = 3/8 }}}
{{{ x = 1/2 }}}
and, also
{{{ x - 1/8 = -3/8 }}}
{{{ x = -1/4 }}}
{{{ r[1] + r[2] = 1/2 - 1/4 }}}
{{{ r[1] + r[2] = 1/4 }}}
OK