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

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

Commented by MJS last updated on 11/Jun/20

$$x^2+xy+y^2\ne \frac{\sqrt{3}}{2}{(x+y)}^2+\frac{1}{2}{(x-y)}^2$$$${\mathrm{there}}^{\prime }\mathrm{s}\mathrm{something}\mathrm{wrong}\mathrm{with}\mathrm{this}\mathrm{question}$$

Commented by MJS last updated on 11/Jun/20

$$x^2+xy+y^2=p{(x+y)}^2+q{(x-y)}^2$$$$\iff$$$$(1-p-q)(x^2+y^2)+(1-2p+2q)xy=0$$$$\Rightarrow$$$$1-p-q=0\wedge 1-2p+2q=0$$$$\Rightarrow$$$$p=\frac{3}{4}\wedge q=\frac{1}{4}$$$$\Rightarrow$$$${\left(\frac{\sqrt{3}}{2}(x+y)\right)}^2+{\left(\frac{1}{2}(x-y)\right)}^2=x^2+xy+y^2$$

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