Cubic equation with more than 3 solutions

Algebra Level 4

How many solutions are there for the following equation:

$\large {\left \{ (x+1)^3 \right \} } = x^3$

Note: $\left \{ x \right \}$ is the fractional part of $x$

1 solution

Ossama Ismail
Mar 18, 2017

Here we can see that $0 \leq x < 1.$

And $\left \{ (x +1)^3 \right \} = \left \{ x^3 + 3x^2 + 3x +1 \right \} = x^3$

Which lead to $3x^2 + 3x = \text{integer value} = k \ \ \text{with} \ \ k = 0,1,2,3,4,5.$

$3 x^2 + 3x - k =0 \implies x = \dfrac{-3 + \sqrt{9 + 12 k}}{6}$

Hence, there are 6 solutions for the above equation.

Why k is 0,1,2,3,4,5. ?

Kanta Sharma - 4 years, 1 month ago

Because x is less than 1.

Ossama Ismail - 4 years, 1 month ago