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.

Source of the problem

Singapore Math Olympiad 2006 (Senior Section – Problem 23)

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

Start with hints

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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