Problema Solution
find two numbers such that their sum is 20 and the sum of their squares is as small as possible.
Answer provided by our tutors
let 'x' and 'y' be the two numbers
their sum is 20
x + y = 20 => y = 20 - x
the sum of their squares is as small as possible
the sum of their square is x^2 + y^2 = x^2 + (20 - x)^2
that is we need to find minimum for the function
f(x) = x^2 + (20 - x)^2
by simplifying the expression we get
x^2 + (20 - x)^2 = 2x^2 - 40x + 400
click here to see the step by step simplifying of the expression
f(x) = 2x^2 - 40x + 400 is a parabolic function
since the quotient before x^2 is positive 2>0 the parabolic function has minimum in its vertex
f min = c - (b^2/4a) where a = 2, b = -40, c = 400
f min = 400 - (-40)^2/(4*2)
f min = 200
this can be also seen in the graph of the function
click here to see the graph of the function
lets find x if we know that f(x) = 200
2x^2 - 40x + 400 = 200
by solving the quadratic equation we find
x = 10
y = 20 - 10 = 10
y = 10
click here to see the step by step solution of the quadratic equation
the numbers are 10 and 10.