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How might I find linear combinations $$\begin{align*} A&=a_1x+a_2y+a_3z\\ B&=b_1x+b_2y+b_3z\\ C&=c_1x+c_2y+c_3z \end{align*}$$

Such that I can transform the two polynomials

$$2x^2+3y^2-2yz+3z^2\text{ and }x^2+6xy+3y^2+2yz-6zx+3z^2$$

into

$A^2+B^2+C^2$ and $\alpha A^2+\beta B^2+\gamma C^2$ respectively for some $\alpha, \beta,\gamma\in \mathbb R$?

I think I should be completing the square, but I can't see how to.

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    So, the first equation is equal to (A^2 +B^2+C^2) and the second to the multiple of it? And are you allowed matrix algebra to solve systems of linear equations?2011-12-04
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    @DrewChristianson: Yes and no. The first equation is indeed equal to $A^2 +B^2+C^2$ but the second one is not necessarily a multiple of it since $\alpha, \beta, \gamma$ are not necessarily equal. Any method of solving it would be appreciated!2011-12-05
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    I think this is the standard linearly algebraic algebra course(Simultaneous Diagonalization); maybe the tag could be changed?2011-12-06

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