Question : If $a + b + c = 10, a^2 + b^2 + c^2 = 38$, what is the value of $(a - b)^2 + (b - c)^2 + (c - a)^2?$
Option 1: 15
Option 2: 12
Option 3: 14
Option 4: 13
Correct Answer: 14
Solution : Given: $a + b + c = 10, a^2 + b^2 + c^2 = 38$ So, $(a+b+c)^2=10^2$ ⇒ $a^2+b^2+c^2+2(ab+bc+ca)=100$ ⇒ $38+2(ab+bc+ca)=100$ ⇒ $2(ab+bc+ca)=62$ Now, $(a - b)^2 + (b - c)^2 + (c - a)^2$ = $2(a^2+b^2+c^2)-2(ab+bc+ca)$ = $(2 × 38) - 62$ = $76 -62$ = $14$ Hence, the correct answer is 14.
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