Solving for X

[TangoFoxtrotGolf]

X^5 - 2X^3 - 2X = 0

I factored X out of the left side,

X(X^4 - 2X^2 - 2) = 0

So one solution is X = 0, which obviously checks in the original equation.

That leaves

X^4 - 2X^2 - 2 = 0,

And I have no idea what to do next

 

[mmm4444bot]

Hi TFG:

Recognize that X^4 - 2X^2 - 2 is quadratic "in form".

To see this, make a substitution for X^2.

X^2 = u

Then, u^2 = X^4.

u^2 - 2u - 2 = 0

Can you solve this equation for u?

If you can, then you can find X because you know that your values for u equal x^2.

Please show whatever work you can try, if you would like more help with this exercise.

 

[Loren]

One approach is to let u=x^2 and then you have...
u^2 - 2u - 2 = 0

Use the quadratic formula to solve for u. Once you have the value of u, put it back into the equation u = x^2 and solve for x.

 

[TangoFoxtrotGolf]

mmm4bot and Loren,

Yep that worked perfectly.

So simple. Substitution works in different ways.

In fact, I think my author was hinting at this with an even more complex earlier exercise, and I did not get it.

Thanks a lot.