Question 194616
let n=2m -30
and  n^2 + m^2 = min
,
(2m-30)^2 +m^2 =0
,
4m^2 -120m +90 +m^2 =0
,
3m^2 -120 m +90 =0
3(m^2 -40m +30 ) =0
,
use  quadratic  eqn ,  a=3, b=-40,c=30
,
m=  (  -(-40) +/- sqrt( -(-40) - 4 (1)(30) ) / 2(1)
m = (40 +/- sqrt (1600-120) )/2
m= (40 +/- 38.47) /2  =  39.2 ,,,, or  .765
,
subst  for  n,   
n= 2m-30 = 2*39.2 -30 = 48.4
or  n= 2* .765 -30 = 28.47
,
pairs  are  m= 39.2,  n=48.4
,
,,,,,,,,m=.765,  n= 28.47
,
,
problem  asks  that  sum  of  squares  is  a  minimum,  therefore  use, m=.765, n=28.47