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Show that the curve $y^2 = x^3 + 2x^2$ has a double point. Find all rational points on this curve.

By implicit differentiation of $x$, $-3x^2 - 4x$ vanishes iff $x = -4/3$ and $0$. By implicit differentiation of $y$, $2y$ vanishes iff $y = 0$.

Taking the second derivative, I got $-6x-4$ and then using the point on the curve $(0,0)$ I got $-4$. Is this my double point?

Thanks for any help!

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    I'd like to give you some advice about the site: **To get the best possible answers, you should explain what your thoughts on the problem are so far**. That way, people won't tell you things you already know, and they can write answers at an appropriate level; also, people are much more willing to help you if you show that you've tried the problem yourself. Also, many would consider your post rude because it is a command ("Show...", "Find..."), not a request for help, so please consider rewriting it.2012-07-19
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    I apologize I am new to this. I have been trying to figure out how to get started on this problem for a while, curves and geometry have always been confusing to me. The book I am using uses more words to teach than examples and I am a visual learner. Any help on how I would go about finding double and rational points on a curve would be greatly appreciated.2012-07-19
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    For the first question, you can perform implicit differentiation on your Cartesian equation; you should find that there is a value of $x$ and $y$ that would make your derivative indeterminate. That is your double point.2012-07-19

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