A simple cubic equation sort of looks like a snake meandering through the origin. Adding other terms changes the way it meanders, which often results in different intersections it makes with the axes.

Our challenge today is to analyze the above equation find out exactly how many solutions it has.

Now are you ready? Let’s dive in.

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Solution

Instead of bringing the 3000 to the left-hand-side, we can factorize the expression on the left.

The next step is to make use of the difference of squares formula.

Because the coefficient of the cubic is positive, the shape must go from negative to positive.

The roots are 0, -sqrt(300) and sqrt(300).

So far so good. But what we are really interested in is how many intersections this makes with the line y = 3000.

Either there’s one one intersection like this.

Two intersections like this.

Or three intersections like that.

The way we can figure this out is by finding the local maximum of the peak on the left.

Using the derivative, we find that the curve achieves it local maximum at x = 10.

Let’s now substitute x = -10 into our equation to find the local maximum point.

And that’s 2000.

So the maximum point must be (-10, 2000), which is well below the line y = 3000.

Therefore, there is only one solution to the equation.