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Question Number 98135 by Tony Lin last updated on 11/Jun/20

h o w t o s p l i t x^{2} + x y + y^{2} i n t o

\frac{\sqrt{3}}{2} {(x + y)}^{2} + \frac{1}{2} {(x - y)}^{2} ?

Commented by MJS last updated on 11/Jun/20

x^{2} + x y + y^{2} \neq \frac{\sqrt{3}}{2} {(x + y)}^{2} + \frac{1}{2} {(x - y)}^{2}

{there}^{'} s something wrong with this question

Commented by MJS last updated on 11/Jun/20

x^{2} + x y + y^{2} = p {(x + y)}^{2} + q {(x - y)}^{2}

\Leftrightarrow

(1 - p - q) (x^{2} + y^{2}) + (1 - 2 p + 2 q) x y = 0

\Rightarrow

1 - p - q = 0 \land 1 - 2 p + 2 q = 0

\Rightarrow

p = \frac{3}{4} \land q = \frac{1}{4}

\Rightarrow

{(\frac{\sqrt{3}}{2} (x + y))}^{2} + {(\frac{1}{2} (x - y))}^{2} = x^{2} + x y + y^{2}

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