If a + b + c = 10 ; $a^2 + b^2 + c^2 = 38$, what is the value of $(a − b)^2 + (b − c)^2 + (c − a)^2$?

14

If a + b + c = 10 ; $a^2 + b^2 + c^2 = 38$

what is the value of $(a − b)^2 + (b − c)^2 + (c − a)^2$

If the number of equations are less than the number of variables then we can put the extra variables according to our choice = 

So here two equations given and three variables are present so put c = 0

If a + b = 10 

$a^2 + b^2  = 38$

then,

( a + b )² = a² + b² + 2ab

(10 )² = 38 + 2ab

2ab = 62

ab = 31

$(a − b)^2 + (b )^2 + (− a)^2$ = a² + b² - 2ab + b² + a²

= 38 + 38 -2 × 31 

= 76 - 62 = 14