Two numbers have a difference of 20. How do you find the numbers if the sum of their squares is a minimum?

2 Answers
Dec 9, 2016

#-10,10#

Explanation:

Two numbers #n,m# such that #n-m=20#

The sum of their squares is given by

#S=n^2+m^2# but #m = n-20# so

#S=n^2+(n-20)^2=2n^2-40n+400#

As we can see, #S(n)# is a parabola with a minimum at

#d/(dn)S(n_0) = 4n_0-40=0# or at #n_0 = 10#

The numbers are

#n=10, m=n-20=-10#

Dec 9, 2016

10 and -10

Solved without Calculus.

Explanation:

In Cesareo’s answer #d/(dn)S(n_0)# is Calculus. Let’s see if we can solve this without calculus.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(magenta)("Let the first number be "x)#
Let the second number be #x+20#

Set #" "y=x^2+(x+20)^2#

#y=x^2+x^2+40x+400#

#y=2x^2+40x+400 larr" "y" is the sum of their squares"#

#color(red)("So we need to find the value of x that gives the minimum value")# #color(red)("of "y)#

This equation is a quadratic and as the #x^2# term is positive then it its general shape is of form #uu#. Thus the vertex is the minimum value for #y#

Write as #y=2(x^2+20x)+400#

What follows is part of the process for completing the square.

Consider the 20 from #20x#

#color(magenta)("Then the first number is: "x_("vertex")=(-1/2)xx20= -10)#

Thus the first number is # x=-10#
The second number is #" "x+20 =-10+20 =10#

#" "color(green)(bar(ul(|color(white)(2/2)"The two numbers are: -10 and 10 "|)))#

Tony B