Question 313304
Let's call the bigger integer {{{x}}} and the smaller integer  {{{y}}}. 

The difference is six. In an equation, this means {{{ x - y = 6 }}}. Another way to write this is: {{{ y = x - 6 }}}. 

if the smaller {{{y}}} is added to the square of the larger {{{x^2}}} the sum is 84: this means {{{ y + x^2 = 84 }}}. 

Let's substitute our expression for {{{ y }}} into this equation. Then we have 

{{{ y + x^2 = ( x - 6) + x^2 = 84 }}}

so subtracting 84 from both sides gives:  {{{ x^2 + x - 90 = 0 }}}. 

Now we can factor: {{{ x^2 + x - 90 = (x - 9) * (x + 10) = 0}}}. Therefore, there are two solutions: {{{ x = 9}}} or {{{x = -10}}}. Since the integer must be positive, we know our answer is {{{ x = 9}}}. 

Now we can find {{{y}}} from the fact that the difference is 6: {{{ x - y = 9 - y = 6}}} implies that {{{y = 3}}}. 

So we have the solution x = 9, y = 3.