(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

For any real numbers a,b,c,d, prove that

[tex]\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)\geq\left(ac+bd\right)^{2}[/tex]

2. The attempt at a solution

I use the triangle inequality to show that

[tex]\left|ac+bd\right|^{2}\leq\left(\left|ac\right|+\left|bd\right|\right)^{2}[/tex]

[tex]\left(ac\right)^{2}+\left(bd\right)^{2}+2abcd\leq\left(ac\right)^{2}+\left(bd\right)^{2}+2\left|ac\right|\left|bd\right|[/tex]

But I'm not sure how to compare it with the final result. Can you assume that

[tex]2\left|ab\right|\left|cd\right|\leq\left(bc\right)^{2}+\left(ad\right)^{2}[/tex]

I know the answer is probably no, and you have to take cases of a<b<c<d, a<c<b<d, etc. But that's 24 cases to consider! And while some are analogous to others, it doesn't cut it by more than half.

Am I even remotely on the right track? Any help or hint would be appreciated.

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# Homework Help: Proving an inequality (not induction)

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