Question 1062169
Let {{{ n }}} = the unknown number
{{{ n^2/2 - n = 11 }}}
Multiply both sides by {{{ 2 }}}
{{{ n^2 - 2n = 22 }}}
Complete the square
{{{ n^2 - 2n + (2/2)^2 = 22 + (2/2)^2 }}}
{{{ n^2 - 2n + 1 = 23 }}}
{{{ ( n - 1 )^2 = (sqrt(23 ))^2 }}}
Take the square root of both sides
{{{ n - 1 = sqrt(23) }}}
{{{ n = 1 + sqrt(23) }}}
and, taking the negative square root of {{{ 23 }}},
{{{ n - 1 = -sqrt(23) }}}
{{{ n = 1 - sqrt(23) }}}
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Approximate values:
{{{ n = 1 + 4.7958 }}}
{{{ n = 5.7958 }}}
and
{{{ n = 1 - 4.7958 }}}
{{{ n = -3.7958 }}}
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Here's the plot:

{{{ graph( 400, 400, -10, 10, -30, 5, x^2 - 2x - 22 ) }}}