Two numbers have a difference of 20. Find the numbers if the sum of their squares is a minimum.

Question

*Please dont use calculus or functions*

Please help, i cant get the answer! Thank you in advance

Answer 1

a - b = 20 ⇒ a = b + 20

a² + b² = minimum

a² + b² = (b + 20)² + b² = (b² + 40b + 400) + b² = 2b² + 40b + 400

Simplify a² + b² by dividing by 2.

(a² + b²)/2 = b² + 20b + 200

Note the expression on the right represents a parabola that opens up, so its vertex is a minimum. Find the vertex by completing the square.

b² + 20b + 200 = (b² + 20b + 20²/4) + (200 - 20²/4) = (b² + 20b + 100) + 100

b² + 20b + 200 = (b + 10)² + 100

This has the form

(b - h)² + k,

where the vertex is (h,k).

h = -10

h is the value of b that minimizes (b - h)² + k, so

b = -10

a = b + 20 = -10 + 20 = 10