SOLUTION: If x+y+z=7 and x^2+y^2+z^2=19, then what is the arithmetic mean (average) of the three products xy, yz, and zx?

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons -> SOLUTION: If x+y+z=7 and x^2+y^2+z^2=19, then what is the arithmetic mean (average) of the three products xy, yz, and zx?       Log On


   

Question 1023639: If x+y+z=7 and x^2+y^2+z^2=19, then what is the arithmetic mean (average) of the three products xy, yz, and zx?
Answer by ikleyn(52754) About Me  (Show Source):
You can put this solution on YOUR website!
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If x+y+z=7 and x^2+y^2+z^2=19, then what is the arithmetic mean (average) of the three products xy, yz, and zx?
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The key to solving this problem is this identity

%28x+%2B+y+%2B+z%29%5E2 = x%5E2+%2B+y%5E2+%2B+z%5E2 + 2%2A%28xy+%2B+xz+%2B+yz%29.

It implies that 

xy+%2B+xz+%2B+yz = %281%2F2%29%2A%287%5E2-19%29 = 15.

Then %281%2F3%29%2A%28xy+%2B+xz+%2B+yz%29 = 15%2F3 = 5.

It is your answer.