perplexus.info :: Just Math : Rooting for Roots

Rooting for Roots (Posted on 2015-02-04)

Find the possible range of a real constant m such that the equation:
x² - 2*x*floor(x) + x = m has at least two non-negative roots.

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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solution

| Comment 1 of 2

From the graph of the LHS (click to see), m can be from 0 to 2, but care must be taken at the end points and the discontinuity in the LHS.

At x=0 the LHS is 0.

At x=1 the LHS is 0.

At x = 2 the LHS is -2.

Therefore 0 <= m < 2, as m=2 results in only one solution.

For negative m, one of the solutions will be negative. Above 2 and there will be no real solutions.

Corrected value of LHS for x=2, per Jer's comment.

Edited on February 5, 2015, 3:24 pm

Posted by Charlie on 2015-02-04 15:25:24

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