Solving Quartic Equation Could someone please explain how to solve this : $x^4+3x^3-6x^2+16x+56=0$ - not the answer only, but a step-by-step solution.                                                        
 A: Quartic equation has a number of different ways to be solved.
One is to write it as a function and to find it's zeroes using Newton's Method or some other zero-approximatization methods.
Another way is to use resolvents(radicals) to solve it. If you're begginer you'd have a little bit harder time with this, but the routine of solving quartic equation will come with time.
Also you could use Ferrari's method to transform this equation to a equation of third degree and then solve it using Cardano's method. This is pretty simular to resolvent solution.
If a quartic equation has a rational solution, then it must be factor of the last coefficient. So you can check all its factor. If you find one then you can factorize it and reduce it to third degree polynomial, which are much easier to solve. Also you can repeat the process again and hopefully bring it to a second degree polynomial. This method is called Rational Root Theorem.
A: Since the coefficient of the fourth power term is 1, let A=3, B=-6, C=16, and D=56.  These are just the remaining coefficients and constant term.  Now transform the quartic into the equivalent resolvent cubic
Y^3 - By^2 + (AC-4D)Y - A^2D + 4BD - C^2 = 0
Using the above values of A, B, C, and D, you should find that this reduces to Y^3 + 6Y^2 - 176Y - 2104 = 0.  Now, we need to find a solution to this cubic.  
So for the time being, redefine A, B, C, and D to be the coefficients and constant term of this cubic...A is 1, B is 6, C is -176 and D is -2104.  Calculate (3C/A) - (B^2/A^2) and divide the difference by 3.  Call the result F.  In this case F = -188.  Now calculate (2B^3/A^3) - (9BC/A^2) + (27D/A) and divide the result by 27.  Call the result G.  Here, G is -1736.  Now calculate (G^2/4) + (F^3/27) and call the result H.  Here, H = 507325.0370.
Since H is positive, the cubic has only one real root.  Define R as (-G/2) + sqrt(H).  Define S as the cube root of R.  Define T as (-G/2) - sqrt(H) and U as the cube root of T.  Here, R = 1580.267532, S = 11.64779019, T = 155.7324681, and U = 5.38013354.  A solution to the resolvent cubic is now given by
X1 = (S+U) - (B/3A), or approximately 15.02792373.
Now go back to the values of A, B, C, and D for the quartic equation and define Y as the root of the resolvent cubic we just found. Take the square root of (A^2/4 - B + Y) and call it R.
Now, calculate 3A^2/4 - R^2 - 2B + (4AB - 8C - A^3)/4R.  Take the square root of this and call it D.  Take 3A^2/4 - R^2 - 2B - (4AB - 8C - A^3)/4R, then find the square root of this and call it E.
The four roots of the quartic can now be expressed as:
1) (-A/4) + (R/2) + (D/2)
2) (-A/4) + (R/2) - (D/2)
3) (-A/4) - (R/2) + (E/2)
4) (-A/4) - (R/2) - (E/2)
In this case, they are approximately -1.81751644, -4.50720359, and 1.66236004 +/- 2.01805997i.
Hope this helps!
