To Know Everyone Knows Identities - 4?

Algebra Level 2

What is the Simplified Form of :

a 2 + b 2 + c 2 − 2 a b + 2 b c − 2 c a \color{#20A900}{a^2 + b^2 + c^2 - 2ab + 2bc - 2ca}

( a − b − c ) 2 (a-b-c)^2 ( a − b + c ) 2 (a-b+c)^2 ( a + b − c ) 2 (a+b-c)^2 ( − a + b − c ) 2 (-a+b-c)^2 ( a + b + c ) 2 (a+b+c)^2

Anish Harsha by Anish Harsha , 20, India

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3 solutions

Chew-Seong Cheong
Jul 17, 2015

a 2 + b 2 + c 2 − 2 a b + 2 b c − 2 c a = a 2 + b 2 + c 2 + 2 ( [ − a ] b + b c + c [ − a ] ) = ( [ − a ] + b + c ) 2 But not an option = ( − 1 [ a − b − c ] ) 2 = ( a − b − c ) 2 \begin{aligned} a^2+b^2+c^2 - 2ab + 2bc - 2ca & = a^2+b^2+c^2 +2([-a]b + bc + c[-a]) \\ & = ([-a]+b+c)^2 \quad \color{#D61F06}{\text{But not an option}} \\ & = (-1[a-b-c])^2 \\ & = \boxed{(a-b-c)^2}\end{aligned}

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Oli Hohman
Sep 1, 2015

Shortcut way of solving since it's multiple choice. Notice, that 2 b c is the only positive non-quadratic term. In order to achieve this from multiplying it out, there are 2 possibilities.

Either the 2bc is positive from multiplying then adding b c twice or it's positive from multiplying then adding -b -c twice

The only answer with either of the two possibilities was (a-b-c)^2

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Zaman Ahsan
Aug 15, 2015

easily find that there was (+) sigh before 2bc. so may both of b & c minus or plus! and where there is "a" in 2ab/ 2ca there lays a minus sigh! so "a" never be negative.... as the result is (a-b-c)^2

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