perplexus.info :: Just Math : Cubic Arithmetic Conclusion

Cubic Arithmetic Conclusion (Posted on 2021-11-16)

Determine (with proof) nonzero integer value(s) of Q such that the equation x³-15x²+Q =0 has three distinct real roots forming an arithmetic sequence.

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)

simple calculus solution

| Comment 3 of 4 |

Cubic graphs have 180 degree rotation symmetry about an inflection point. If this point is on the x-axis, the other two roots will be equal distances +/- and so form an arithmetic sequence.

The second derivative of the equation is 6x-30 so the inflection point occurs when x=5.

5^3-15*5^2=-250.

So raise the graph by making Q=250.

Posted by Jer on 2021-11-17 11:26:27

Please log in:

Login:
Password:
Remember me:

Sign up! \| Forgot password

Search:

Search body:

Forums (1)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (6)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:


perplexus dot info