Finding the min value of an expression

[superconduct]

Homework Statement

a,b,c,d are real numbers
ab+bc+cd+da=16
Find the minimum value possible for a²+b²+c²+d²

Homework Equations

ab+bc+cd+da = (a+c)(b+d)
x^2 \ge 0 for some real x

The Attempt at a Solution

1st approach:2(a^2 +b^2 +c^2 +d^2) + 2(ab+bc+cd+da) = (a+b)^2 +(b+c)^2 +(c+d)^2 +(d+a)^2 \ge 0
which gives a^2 +b^2 +c^2 +d^2 \ge -16 that cannot give the min. value.

2nd approach: (a+c)(b+d) =16
By differential calculus, rectangle of fixed area has min. diagonal when it is square,
a+c=b+d=4
a=b=c=d=2
a^2 +b^2 +c^2 +d^2 \ge 2^2+2^2+2^2+2^2 =16

How would you solve this? If you think there is any unclear reasonings in my post or others' replies please do not mind elaborating/criticizing/replacing them for me and others.

 

[Joffan]

Your first approach is good, but try subtracting instead of adding. Then you can demonstrate (as you have) that the lowest answer is achievable.