this may seem very dumb, but for some reason I can't quite remember how to simplify an equation of the following form in order to get $x$ in terms of $y$ or vice-versa
$$ x^2 -2xy = y - y^2 $$
i keep getting stuck at the $2xy$ term
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1Complete the square. $$x^2-2xy+y^2=y$$Is more suggestive – FShrike Feb 09 '23 at 22:48
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what do you mean? I am still left with the $-2x$ if i try and move the 2xy term to the right and factor out the y... – JoeVictor Feb 09 '23 at 22:49
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1Completing the square means you have something like $(x-a)^2+b$ instead of $x^2+\cdots$ – FShrike Feb 09 '23 at 22:50
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1ah gotcha! that edit helped. let me give it a try :) – JoeVictor Feb 09 '23 at 22:50
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1E.g. could you complete the square in $x^2-4x+5$ (to then solve the quadratic $\cdots=0$)? – FShrike Feb 09 '23 at 22:51
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i know that there's a formula for that... looking it up now – JoeVictor Feb 09 '23 at 22:54
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$x = \sqrt{y} + y$ – JoeVictor Feb 09 '23 at 22:59
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my high school math is a bit rusty, thanks so much! – JoeVictor Feb 09 '23 at 22:59
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1I'd say $x=y\pm\sqrt y$. – Gerry Myerson Feb 09 '23 at 23:06
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sure, that is the more correct answer. in this specific question though i know x & y have to be positive though! – JoeVictor Feb 09 '23 at 23:35
1 Answers
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To go about solving this one, remember the property that $(a+b)^2=a^2+2ab+b^2$. This comes from the definition of squaring, to multiply something by itself, so we see that $(a+b)^2=(a+b)(a+b)$. Using FOIL, this evaluates to the equation above.
Next, recall that any negative number squared is positive. That is, $(a-b)^2=a^2-2ab+b^2$. If we replace the $a$ and $b$ with $x$ and $y$, we get that $(x-y)^2=x^2-2xy+y^2$. Now, it would make sense to rearrange the equation by adding $y^2$ to each side, into the equation $x^2−2xy+y^2=y$. Using the property above, this is equivalent to $(x-y)^2=y$.
Following this step, take the square root of each side into $x-y=\pm\sqrt{y}$. the final step is to add $y$, which gives the equality $x=y\pm\sqrt{y}$. Hope this helps!
