# Real Solutions for a Quartic Equation

I am looking for the real solutions for the equation: $$Ax^4+Bx^3+Cx^2+Dx+E=0$$ I followed the wikipedia page on quartics and found the discriminant, P, and Q such that the cases of solutions are:

If ∆ < 0 then the equation has two distinct real roots and two complex conjugate non-real roots.

If ∆ > 0 then either the equation's four roots are all real or none is.

If P < 0 and D < 0 then all four roots are real and distinct.

If P > 0 or D > 0 then there are two pairs of non-real complex conjugate roots. If ∆ = 0 then (and only then) the polynomial has a multiple root. Here are the different cases that can occur: If P < 0 and D < 0 and ∆0 ≠ 0, there is a real double root and two real simple roots.

If D > 0 or (P > 0 and (D ≠ 0 or Q ≠ 0)), there is a real double root and two complex conjugate roots.

If ∆0 = 0 and D ≠ 0, there is a triple root and a simple root, all real.

If D = 0, then:

If P < 0, there are two real double roots.

If P > 0 and Q = 0, there are two complex conjugate double roots.

If ∆0 = 0, all four roots are equal to $-\frac{b}{4a}$

Where $\Delta$, P, Q, D, $\Delta_0$ are as defined in https://en.wikipedia.org/wiki/Quartic_function

I was curios if someone could point me in the correct direction as I was very confused by the rest of the Wikipedia page.

Once again, I only need the real solutions.

Thanks in advance, Zach Hilman

• The formula for the general quartic (real) equation is a great achievment of some 500 years ago (Ferrari, Cardano, Tartaglia and pals), but the formula is so ugly and cumbersome that almost nobody really studies and learns it by heart. You can check hundreds of papers about this in the web, and also here: mathworld.wolfram.com/QuarticEquation.html – DonAntonio Aug 2 '16 at 23:23
• Besides brute force mentioned to tackle general quartic equations, you may use factor theorem to find out possible linear factors (by trial-and-error). Another trick is to see whether it's reducible to the form of $A(x^2+ax+b)(x^2+cx+d)$ by equating coefficients. – Ng Chung Tak Aug 3 '16 at 1:03