November 18th, 2018, 09:00 AM  #1 
Member Joined: Oct 2012 Posts: 71 Thanks: 0  minimum value
Let a,b,c and d to be a positive real. Such that: (a+b)(c+d)=143 (a+c)(b+d)=150 (a+d)(b+c)=169. Find the min value of a^2+b^2+c^2+d^2? 
November 18th, 2018, 11:20 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,366 Thanks: 1272 
Applying the method of Lagrange Multipliers using 3 constraint equations and with the help of software to churn through the considerable algebra I come up with the only value of the sum of the squares of a,b,c,d that also satisfies the constraints is $a^2 + b^2 + c^2 + d^2 = 214$. Are you familiar with how to use Lagrange Multipliers? There may be some much simpler method. Last edited by romsek; November 18th, 2018 at 11:30 AM. 

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