Is there a number that is exactly 3 more than its cube?

For a given number X, its cube is written as X^3.

Here we want to see if we can find a number that is exactly 3 more than its cube, this means that:

We will see that yes, there exists a number with this property.

Then we can try to solve this to find the number with that given property.

We can rewrite it as:

Now we can solve this for X, as the question asks about existence and not for the exact number, we can graph this the above function:

And if it crosses the x-axis, then at some point we have:

This would imply that there exists a real number that is 3 more than its cube.

The graph can be seen below, and we can see that it intersects the X-axis, then yes, there is a number that is exactly 3 more than its cube.

And by watching the graph, we can see that the number is:

If you want to learn more, you can read:

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step-by-step explanation: