Quadratic and Floor (part 2)

Algebra Level 5

$\large x^2-25\lfloor x\rfloor +b=0$

At most, how many distinct real roots can the above equation has, over the real number $b$ ?

Notation : $\lfloor \cdot \rfloor$ denotes the floor function .

2 4 12 8 10 6 Infinitely many More than 12 but finitely many

1 solution

Chan Lye Lee
Jun 12, 2016

Note that the graph of $y=x^2-25 \lfloor x \rfloor +b$ is formed by lots of curves (which 'closed to' segments of gradient $2x-25$ if $x$ is large) and "bounded" between the graphs of $y=x^2-25x+b$ and $y=x^2-25(x-1)+b$ .

The largest number of intersection points between $y=x^2-25 \lfloor x \rfloor +b$ and $y=0$ is $\boxed{10}$ , from the graph shown, when is closed to vertex of the parabolas.

Quadratic and Floor (part 2)

1 solution

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