# Thread: How do I find x with this polynomial function?

1. ## How do I find x with this polynomial function?

How do I determine the values of x when f(x) = 48 when f(x) = x4 - 5x2 + 4. Please and thank you!

2. ## Re: How do I find x with this polynomial function?

Let $\displaystyle w = x^2$, then solve for w, then take square roots to find x values.

Some polynomials are "secretly" solvable as quadratics. Here are some other eamples:

$\displaystyle 4x^{20}-4x^{10}+1 = 0, \ q^8 + 7q^4 +6 = 0, \ z^{14}-z^7-56 = 0$

3. ## Re: How do I find x with this polynomial function?

$\displaystyle x^4 - 5x^2 + 4 = 48$

$\displaystyle x^4 - 5x^2 = 44$

complete the square ...

$\displaystyle x^4 - 5x^2 + \frac{25}{4} = 44 + \frac{25}{4}$

$\displaystyle \left(x^2 - \frac{5}{2}\right)^2 = \frac{201}{4}$

$\displaystyle x^2 - \frac{5}{2} = \pm \frac{\sqrt{201}}{2}$

assuming you're looking for real roots only ... $\displaystyle x^2 > 0$ for all $\displaystyle x \ne 0$

$\displaystyle x^2 = \frac{5 + \sqrt{201}}{2}$

$\displaystyle x = \pm \sqrt{\frac{5 + \sqrt{201}}{2}}$