Question

Find the value of A=2/a-b + 2/b-c + 2/c-a + [(a-b)²+(b-c)²+(c-a)²]/(a-b)(b-c)(c-a)
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Answer 1

`A = (2)/((a - b)) + (2)/((b - c) ) + (2)/((c - a) ) + ((a-b)²+(b-c)²+(c-a)²)/((a - b)(b - c)(c - a)) \\ \: \: \: = (2(b - c)(c - a) + 2(a - b)(c - a) + 2(a - b)(b - c))/((a - b)(b - c)(c - a)) + ((a-b)²+(b-c)²+(c-a)²)/((a - b)(b - c)(c - a)) \\ \: \: = \frac{2(bc - ab - {c}^(2) + ac) + 2(ac - {a}^(2) - bc + ab) + 2(ab - ac - {b}^(2) + bc) + a² - 2ab + b ² + b² - 2bc+c²+c²-2ac+a²}{(a - b)(b - c)(c - a)} \\ \: \: = \frac{2bc - 2ab - 2{c}^(2) + 2ac+ 2ac - 2 {a}^(2) - 2bc + 2ab+ 2ab - 2ac - 2 {b}^(2) + 2bc+ a² - 2ab + b ² + b² - 2bc+c²+c²-2ac+a²}{(a - b)(b - c)(c - a)} \\ \: \: = (a² + b ² +c² )/((a - b)(b - c)(c - a)) \\ \: \:`

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