# Thread: Solving 4th order polynomial equations

1. ## Solving 4th order polynomial equations

How do you solve for x with a fourth order polynomial?

The problem I'm working with now is asking for the largest possible value of x for the equation 8x^4 - 16x^2 = 4.

I've played around with it some, but can't work out a viable way to solve for x. Any suggestions are appreciated.

2. ## Re: Solving 4th order polynomial equations

Have you considered the Quadratic nature of this particular example? If you don't see it, substitute y = x^2 and it should become obvious.

3. ## Re: Solving 4th order polynomial equations

Originally Posted by TKHunny
Have you considered the Quadratic nature of this particular example? If you don't see it, substitute y = x^2 and it should become obvious.
Hmmm...I'm afraid I don't follow. I factored and tried to adopt it into quadratic form, but it didn't work out. The substitution didn't really open it up for me, but I'm sure I'm missing something simple...

4. ## Re: Solving 4th order polynomial equations

May as well start by cancelling 4: $\displaystyle 2x^4 - 4x^2 -1 =0$

This is equal to $\displaystyle 2(x^2)^2 - 4(x^2) -1 =0$. Taking TKHunny's suggestion and saying that $\displaystyle y=x^2$ then it becomes $\displaystyle 2y^2-4y-1=0$. That is standard quadratic form which can be solved using your favourite method.

Once you have solutions for y take the square root to get your x values.

edit: This equation doesn't factor, use the quadratic formula or complete the square

edit 2: You'll get four solutions although some may be complex

5. ## Re: Solving 4th order polynomial equations

Thanks guys, it worked out well.

I appreciate the help!