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.