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# 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. $(\frac{-1}{\sqrt [3]{2}}, \frac{-3}{\sqrt [3]{4}})$

So the number distinct point will be 1

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#### Math Olympiad Program

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.