Question 663417
Suppose I vary one of the numbers and hold
the other number fixed, but not saying 
right away what it is fixed at.
Let the variable number = {{{ x }}}
Let the fixed number = {{{ a }}}
given:
{{{ x + a = 18.5 }}}
{{{ f(x) = x^2 + a^2 }}}
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{{{ a = 18.5 - x  }}}
{{{ f(x) = x^2 + ( 18.5 - x )^2 }}}
{{{ f(x) = x^2 + 342.25 - 37x + x^2 }}}
{{{ f(x) = 2x^2 - 37x + 342.25 }}} ( just what you got )
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This is in the form {{{ ax^2 + bx + c }}}, so the 
minimum is at {{{ -b/(2a) }}}
{{{ a = 2 }}}
{{{ b = -37 }}}
{{{ -b/(2a) = -(-37) / (2*2) }}}
{{{ -b/(2a) = 37/4 }}}
{{{ 37/4 = 9.25 }}}
and
{{{ a = 18.5 - x  }}}
{{{ a = 18.5 - 9.25 }}}
{{{ a = 9.25 }}}
Both numbers are 9.25
check:
{{{ 2*9.25^2  = 171.125 }}}
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What if the numbers are
9.24 and 9.26
{{{ 9.24^2 + 9.26^2 = 85.3776 + 85.7476 }}}
{{{ 85.3776 + 85.7476 = 171.1252 }}}
This is higher than {{{ 2*9.25^2 }}}
So we have a minimum