 # Understand the problem

Consider the two curves $y=2x^3+6x+1$ and $y=-3/x^2$ in the Cartesian plane. Find the number of distinct points at which these two curves intersect.

#### Singapore Math Olympiad 2006 (Senior Section – Problem 23)

##### Topic
Intersection of curves
5 out 10
##### Suggested Book
Pre College Mathematics

Do you really need a hint? Try it first!

Try to think how to find the intersection points using these two given equation $y=2x^3+6x+1$ and $y=-3/x^{2}$

Try to compare the two values of y e.g. $2x^3+6x+1=-3/x^{2}$ Do we get two factors $2x^3+1=0$ and $x^2+3=0$

At end as we will consider only $2x^3+1=0$ as $x^2+3>0$

Thus we will get the value of x and from there we can find the value of y and we will get the answer. $(\frac{-1}{\sqrt {2}}, \frac{-3}{\sqrt {4}})$

So the number distinct point will be 1

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