Question 451634
We have two numbers x and y.

Their sum is 9 -> {{{x+y=9}}}
Sum of their squares is 101 -> {{{x^2+y^2=101}}}


If we square first equation we get


{{{(x+y)^2=9^2}}}
{{{x^2+2xy+y^2=81}}}


Now we can replace {{{x^2+y^2}}} with 101


{{{2xy+101=81}}}
{{{2xy=-20}}}
{{{xy=-10}}}


From their sum we have that y=9-x, so


{{{x(9-x)=-10}}}, multiply
{{{9x-x^2=-10}}}, move everything on one side
{{{x^2-9x-10=0}}}, now we have quadratic equation and can find solutions for x


{{{x = (-(-9) +- sqrt( (-9)^2-4*1*(-10) ))/(2*1) }}}

We get x = 10 or x = -1.
For x=10 we get y=9-x=9-10=-1.
For x=-1 we get y=9-(-1)=10


So two numbers are 10 and -1.


Check: {{{10+(-1)=9}}}, OK
       {{{10^2+(-1)^2=100+1=101}}}, OK