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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)

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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

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