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If the sum of the squares of nonnegative real numbers $a,b,$ and $c$ is $13$, and $ab + bc + ca = 6$, then what is the sum of $a,b,$ and $c$?

 Jun 24, 2019
 #1
avatar+87 
+1

So, we know that a^2 + b^2 + c^2 = 13.

Also, ab + bc + ca = 6.

 

And we want to find a + b + c. So, we sqaure a + b + c.

That would be a^2 + ab + ac + ba + b^2 + bc + ca + cb + c^2.

We group the squares together to get a^2 + b^2 + c^2, which is 13.

Then we group the ab or bc or ac together.

 

That gets us 2(ab + bc + ac) which is 2*6 which is 12.

Then we add to get 13 + 12 = 25.

 Jun 24, 2019
 #2
avatar+101813 
+1

Nice, Pushy!!!

 

To  get the su of  a + b + c

 

(a + b + c)^2   =   25      [ as Pushy found ]

 

Take the square root

 

a + b + c  =   5

 

cool cool cool

CPhill  Jun 24, 2019
edited by CPhill  Jun 24, 2019
edited by CPhill  Jun 24, 2019

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