Expansion of (a ± b ± c)$$^{2}$$

We will discuss here about the expansion of (a ± b ± c)$$^{2}$$.

(a + b + c)$$^{2}$$ = {a + (b + c)}$$^{2}$$ = a$$^{2}$$ + 2a(b + c) + (b + c)$$^{2}$$

= a$$^{2}$$ + 2ab + 2ac + b$$^{2}$$ + 2bc + c$$^{2}$$

= a$$^{2}$$ + b$$^{2}$$ + c$$^{2}$$ + 2(ab + bc + ca)

= sum of squares of a, b, c + 2(sum of the products of a, b, c taking two at a time}.

Therefore, (a – b + c)$$^{2}$$ = a$$^{2}$$ + b$$^{2}$$ + c$$^{2}$$ + 2(ac – ab – bc)

Similarly for (a – b – c)$$^{2}$$, etc.

Corollaries:

(i) a$$^{2}$$ + b$$^{2}$$ + c$$^{2}$$ = (a + b + c)$$^{2}$$ – 2(ab + bc + ca)

(ii) ab + bc + ca = $$\frac{1}{2}$${(a + b + c)$$^{2}$$ – (a$$^{2}$$ + b$$^{2}$$ + c$$^{2}$$)}

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