Positive roots

Algebra Level 3

$x^6-x^3+6=0$

How many positive roots does the equation above have?

2 solutions

Hi Bye
Nov 4, 2019

Let $y=x^3.$ Then, we have to find all solutions to $y^2-y+6=0,$ which clearly doesn't have any real roots. Thus, the answer is $\boxed{0}.$

Richard Desper
Nov 5, 2019

For $0 < x < 1$ , $x^6 >0$ and $|x^3| < 1$ , thus $x^6 - x^3 + 6 > 5$ .

For $x \geq 1$ , $x^6 \geq x^3$ , thus $x^6 - x^3 + 6 \geq 6$ .

Thus there are no positive roots to $x^6 - x^3 + 6 = 0$ .

Alternatively, complete the square to rewrite $x^6 - x^3 + 6$ as $(x^3 - \frac{1}{2})^2 + 5.75$ , which clearly has a global minimum of $5.75$ .

Richard Desper - 1 year, 7 months ago