Add the following:

$a (a + b), b (b + c), c (a + c) a n d a^{2} + b^{2} + c^{2}$ [2 MARKS]

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Solution

Concept: 1 Mark
Application: 1 Mark

$\Rightarrow a (a + b) + b (b + c) + c (a + c) + a^{2} + b^{2} + c^{2}$

$\Rightarrow a^{2} + a b + b^{2} + b c + c^{2} + c a + a^{2} + b^{2} + c^{2}$

$\Rightarrow 2 a^{2} + 2 b^{2} + 2 c^{2} + a b + b c + c a$

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a (a + b), b (b + c), c (a + c)

a^{2} + b^{2} + c^{2}

Add the following:

$a (a + b), b (b + c), c (a + c) a n d a^{2} + b^{2} + c^{2}$ [2 MARKS]

\frac{a {(b - c)}^{2}}{(c - a) (a - b)} + \frac{b {(c - a)}^{2}}{(a - b) (b - c)} + \frac{c {(a - b)}^{2}}{(b - c) (c - a)} = a + b + c

a^{2} + b^{2} + c^{2} - a b - b c - c a = \frac{1}{2} [{(b - c)}^{2} + {(c - a)}^{2} + {(a - b)}^{2}]

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