The difference between two numbers is 4. The sum of their squares is a minimum. What are the numbers? so far I have a - b = 4 and a^2 + b^2 = c (c is a minimum) then I isolated for a and plugged it into the second equation a = 4 + b then (4+b)^2 + b^2 = c this becomes (4+b)(4+b) + b^2 = c which equals 16 + 8b + 2b^2 = c then I rearranged it to 2b^2 + 8b + 16 = c and then I factored out the 2 which gives us 2(b^2 + 4b + 8) = 0 I'm having trouble factoring this by using decomposition. Any help would be appreciated.
The difference between the two numbers is 4. The sum of their squares is a minimum. What are the numbers?
a - b = 4
2 - (-2) = 4
22 + (-2)2 = 8
+128475 a - b = 4
b = a - 4
a^2 + b^2
a^2 + ( a - 4)^2
a^2 + a^2 - 6a + 16
2a^2 - 8a + 16
a^2 - 4a + 8 take the derivative and set to 0
2a - 4 = 0
2a = 4
a = 2
b = a - 4 = 2 - 4 = -2
a = 2
b = -2 produce a min
