I'm having some trouble solving for $x$ in the following quartic equation.
$$ 0=6x^4+2x^3+4x^2-6x-3 $$
Do you have any suggestions on how I should go about solving this equation? I tried using the rational root theorem and depressing the quartic, but I became very quickly lost.
I'd really appreciate your help! Thank you so much!
Except in few cases (obvious roots, possible factorization), I do not know many people solving quatic equations with radicals.
If you are patient, compute the discriminant; in this case $\Delta=-42288$ tells that "the equation has two distinct real roots and two complex conjugate non-real roots". Good news !
Plot th function and you will notice that the real roots are close to $-0.4$ and $0.9$. From these guesses, start Newton method which will converge lika a charm.
$$\left( \begin{array}{cc} n & x_n \\ 0 & -0.4000000 \\ 1 & -0.3932897 \\ 2 & -0.3932557 \end{array} \right)$$
$$\left( \begin{array}{cc} n & x_n \\ 0 & 0.9000000 \\ 1 & 0.8900408 \\ 2 & 0.8898740 \\ 3 & 0.8898739 \end{array} \right)$$
