I tried to find $ text {k} $ such as $: $

$$ text {discriminant} left [ text{discriminant}left [ z^{,2}left ( z^{,2}- 1 right )^{,2}left ( z^{,2}+ 1 right )+ left ( y^{,2}- z^{,2} right )left ( y^{,4}+ z^{,6}- z^{,4}- z^{,2} right )- text{k}left ( z^{,4}- 1 right )z^{,2}left ( y^{,2}- z^{,2} right ),,y right ] ,, z right]= 0 $$

or $: $

$$ text {discriminant} left [ text{discriminant}left [ z^{,2}left ( z^{,2}- 1 right )^{,2}left ( z^{,2}+ 1 right )+ left ( y^{,2}- z^{,2} right )left ( y^{,4}+ z^{,6}- z^{,4}- z^{,2} right )- text{k}left ( z^{,4}- 1 right )z^{,2}left ( y^{,2}- z^{,2} right ),,z right ] ,, y right]= 0 $$

Then I used **Wolfram Alpha** but it can$ & $t bring a polymonial of $ text {k} $ $, $ see $: $

https://www.wolframalpha.com/input/?i=discriminant%5By%5E6%2By%5E2z%5E6%2Bz%5E2-y%5E2z%5E2 (1% 2By% 5E2% 2Bz% 5E2) -k ( 1% 2Bz% 5E2) z% 5E2 (y% 2Bz) (yz) (z-1) (z% 2B1)% 5D, z% 5D

$$ text {discriminant} left [ z^{,2}left ( z^{,2}- 1 right )^{,2}left ( z^{,2}+ 1 right )+ left ( y^{,2}- z^{,2} right )left ( y^{,4}+ z^{,6}- z^{,4}- z^{,2} right )- text{k}left ( z^{,4}- 1 right )z^{,2}left ( y^{,2}- z^{,2} right ),,z right ]= 0 $$

https://www.wolframalpha.com/input/?i=discriminant%5By%5E6%2By%5E2z%5E6%2Bz%5E2-y%5E2z%5E2 (1% 2By% 5E2% 2Bz% 5E2) -k ( 1% 2Bz% 5E2) z% 5E2 (y% 2Bz) (yz) (z-1) (z% 2B1)% 5D, y% 5D

$$ text {discriminant} left [ z^{,2}left ( z^{,2}- 1 right )^{,2}left ( z^{,2}+ 1 right )+ left ( y^{,2}- z^{,2} right )left ( y^{,4}+ z^{,6}- z^{,4}- z^{,2} right )- text{k}left ( z^{,4}- 1 right )z^{,2}left ( y^{,2}- z^{,2} right ),,y right ]= 0 $$

But $: $

https://www.wolframalpha.com/input/?i=discriminant%5Bdiscriminant%5By%5E6%2By%5E6%2By%5E6%2Bz%5EB2%2%%%%%%%%%%>. k (1% 2Bz% 5E2) z% 5E2 (y% 2Bz) (yz) (z-1) (z% 2B1)% 5D, y% 5D, z% 5D

I've used these to solve an inequality $ ($ his original idea is a **IMO** problem $) $ $. $ How can I get this $ text {k} $ $? $ I need help $! $

Good luck to everyone $! $