Question 613358
Express the 3 consecutive integers as {{{ n }}}, {{{ n + 1 }}}, {{{ n + 2 }}}
{{{ n }}} represents the agr of the youngest child
given:
 {{{ n^2 = 8*( n + 2 ) + 4 }}}
---------------------
{{{ n^2 = 8n + 16 + 4 }}}
{{{ n^2 - 8n - 20 = 0 }}}
I can solve completing the square
{{{ n^2 - 8n = 20 }}}
take 1/2 of the coefficient of the {{{ n }}} term, square it,
and add it to both sides
{{{ n^2 - 8n + ( -8/2 )^2 = 20 + ( -8/2 )^2 }}}
{{{ n^2 - 8n + 16 = 20 + 16 }}}
{{{ n^2 - 8n + 16 = 36 }}}
Both sides are now perfect squares:
{{{ ( n - 4 )^2 = 6^2 }}}
Take the square root of both sides
{{{ n - 4 = 6 }}}
{{{ n = 10 }}}
I can ignore the negative square root, since it gives me
a negative age: {{{ n - 4 = -6 }}}
{{{ n = 10 }}}
{{{ n +1 = 11 }}}
{{{ n + 2 = 12 }}}
The ages of the 3 children is 10, 11, and 12
check:
{{{ n^2 = 8*( n + 2 ) + 4 }}}
{{{ 10^2 = 8*( 10 + 2 ) + 4 }}}
{{{ 100 = 8*12 + 4 }}}
{{{ 100 = 96 + 4 }}}
{{{ 100 = 100 }}}
OK