# Thread: Finding the co-ords where y = 0..

1. ## Finding the co-ords where y = 0..

$\displaystyle y = x^{1/3} - x^2$

How do I solve this equation to find the roots. It's not a quadratic equation because it involves cubic root of x.

2. Hi there!

Originally Posted by struck
$\displaystyle y = x^{1/3} - x^2$

How do I solve this equation to find the roots. It's not a quadratic equation because it involves cubic root of x.
I would try:

$\displaystyle 0 = x^{1/3} - x^2$

$\displaystyle x^2 = x^{1/3}$

next step: (both sides)^3

$\displaystyle (x^2)^3 = x^1$

$\displaystyle x^6 = x^1$

$\displaystyle 0 = x^6 - x^1$

$\displaystyle 0 = x (x^5 - 1)$

=> x = 0 or x=1 (solving $\displaystyle x^5 = 1$ leads to x = 1)

3. Originally Posted by struck
$\displaystyle y = x^{1/3} - x^2$

How do I solve this equation to find the roots.
$\displaystyle 0 = x^{\frac{1}{3}} - x^2$

$\displaystyle 0 = x^{\frac{1}{3}}\left(1 - x^{\frac{5}{3}}\right)$

$\displaystyle x^{\frac{1}{3}} = 0$ ... $\displaystyle x = 0$

$\displaystyle x^{\frac{5}{3}} = 1$ ... $\displaystyle x = 1$