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I have this equation $3-x-x^2$ which I want to convert into the form $a^2 - y^2$, so that then I can suppose $y$ = $a\sin\theta$, differentiate and then integrate it. But I can't convert that equation into this $a^2 - y^2$ form.

P.S. this is a homework question. My question here is not to solve this for me, but tell me how to do such (if there exists a formula) so that I may learn and prosper :)

EDIT: $a^2 - y^2$, in this equation $y$ may be anything which involves $x$ e.g. $(x-2)^2$

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@Tux: I added the [homework] tag, since you say it is a homework question, and I've changed your $x$ into a $y$ so that it is clear that you are allowing functions of $x$. –  Arturo Magidin Dec 17 '10 at 19:40
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3 Answers

What you are trying to do is called completing the square.

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HINT $\rm\quad A^2 - (X-B)^2\ =\ 3 - X - X^2\ \Rightarrow\ B = \ldots\ \Rightarrow\ A = \ldots\ $ (expand and compare coeff's)

Generally if you want to check that some specific polynomial is an instance of some general polynomial form you may treat the coefficients of the general form as variables or indeterminates (a.k.a. "undetermined coefficients"). With luck, the system of equations obtained by comparing coefficients might be easily solvable (e.g. it might be triangular). This is sometimes referred to as the method of undetermined coefficients and it works more generally for objects that are uniquely determined by their "coefficients", e.g. vectors, matrices, (formal) power series, radix notation, partial fraction expansions, etc.

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The point of Bill's hint is that if two polynomials are equal, then their coefficients are equal. –  Andres Caicedo Dec 17 '10 at 19:50
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@Andres: Thanks for the clarifying tip - I've elaborated a bit (perhaps too much). –  Bill Dubuque Dec 17 '10 at 20:20
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Hint. step one is to change the 3 into 3.25 minus 1/4

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