MHB Find the greatest positive integer

[anemone]

Find the greatest positive integer $x$ such that $x^3+4x^2-15x-18$ is the cube of an integer.

 

[kaliprasad]

Find the greatest positive integer $x$ such that $x^3+4x^2-15x-18$ is the cube of an integer.

I had a slice of luck here

Spoiler

x must satisfy

$x^3 + 4x^2 - 15 x - 18 \le (x+1)^3$

or $x^2-18x - 19 \le 0$
so x = 19 makes the RHS 0

so x = 19 is the ans because we get a perfect square ( things would have been different had we not got integer)

 

[anemone]

Well done, kaliprasad!(Yes)

The trick is to recognize that the given expression is less than or equal to $(x+1)^3$.

Thanks for participating as well, my friend!:)