Solving Quartic Equations

Quartic equations have the general form:

a X4   +   bX 3   +   cX2  +   dX   +   e   =   0

Example # 1
Quartic Equation With 4 Real Roots
3X4   + 6X3   - 123X2   - 126X + 1,080 = 0

Quartic equations are solved in several steps. First, we simplify the equation by dividing all terms by 'a', so the equation then becomes:

X4   +   2X 3   -   41X2  -   42X   +   360   =   0

Where   a  =  1   b  =  2   c  =  -41   d   =   -42   and   e   =   360

Next we define the variable 'f':

f = c - (3b2/8)

"Plugging in" the numbers from the above equation, we get:

f = -41 - (3*2*2/8)
f = -42.5

Next we define 'g':

g = d + (b3 / 8) - (b*c/2)
"Plugging in" the numbers:
g = -42 + (8/8) - (2 * -41 / 2)
g = 0

Next, we define 'h':

h = e - (3*b4/256) + (b 2 * c/16) - ( b*d/4)

Plugging in the numbers:

h = 370.5625

Next, we "plug" the numbers 'f', 'g' and 'h' into the following cubic equation:

Y3 + (f/2)*Y2 + ((f2 -4*h)/16)*Y -g2/64 = 0

Y3 -21.25*Y2 + (1,806.25 -4 * 370.5625 )/16*Y -02/64 = 0

Y3 -21.25*Y2 + (1,806.25 -1,482.25)/16*Y -02/64 = 0

Y3 -21.25*Y2 + 20.25*Y -0 = 0

Next, we solve this cubic equation by using the method located at solving cubic equations   OR (much easier) using the

CUBIC   EQUATION   CALCULATOR.

And the 3 roots of the equation are:

Y1=  20.25     Y2=  0     Y3=  1

Let 'p' and 'q' be the square roots of ANY 2 non-zero roots (Y1 Y2 or Y3).

p=SqRoot(20.25) = 4.5

q=SqRoot(1) = 1

r= -g/(8*pq) = 0

s= b/(4*a) = 6/(4*3) = 0.5

Then the four roots of the quartic equation are:

X1= p + q + r -s = 4.5 + 1 + 0 - .5 = 5
X2= p - q - r -s = 4.5 - 1 - 0 - .5 = 3
X3= -p + q - r -s = -4.5 + 1 - 0 - .5 = -4
X4= -p - q + r -s = -4.5 - 1 + 0 - .5 = -6


Example # 2
Quartic Equation With 2 Real and 2 Complex Roots
-20X4   + 5X3   + 17X2   - 29X + 87 = 0

Simplify the equation by dividing all terms by 'a', so the equation then becomes:

X4   -   .25X 3   -   .85X2  +   1.45X   -   4.35   =   0

Where   a  =  1   b  =  -.25   c  =  -.85   d   =   +1.45   and   e   =   -4.35


f = c - (3b2/8)

f = -.8734375

g = d + (b3 / 8) - (b*c/2)

g = 1.341796875

h = e - (3*b4/256) + (b2 * c/16) - ( b*d/4)

h = -4.262741088867187

Next, we "plug" the numbers 'f', 'g' and 'h' into the following cubic equation:

Y3 + (f/2)*Y2 + ((f2 -4*h)/16)*Y -g2/64 = 0

Y3 -0.436718750000*Y2 + 1.113366088867*Y -0.028131544590 = 0

Next, we obtain the 3 roots of this cubic equation by going to the:

CUBIC   EQUATION   CALCULATOR.

and the roots are:

Y1=  0.0255074144632
Y2=  0.2056056677683 + i* 1.029856038619
Y3=  0.2056056677683 - i* 1.029856038619

Let 'p' and 'q' be the square roots of ANY 2 non-zero roots (Y1 Y2 or Y3).

Whenever we have 1 Real Root and 2 complex roots, we ALWAYS choose the 2 complex roots.
Find the square roots by going to the: Complex Number Calculator.

p=SqRoot(Y2) = 0.7923967592303 + i* 0.6498360995438

q=SqRoot(Y3) = 0.7923967592303 - i* 0.6498360995438

r= -g/(8*pq) = -1.341796875000/(8*1.0501795803089815)
=-0.159710408124

s= b/(4*a) = 5/(4*-20) = -.0625

Then the four roots of the quartic equation are:

X1= p + q + r -s =
+ (0.7923967592303 + i* 0.6498360995438)
+ (0.7923967592303 - i* 0.6498360995438)
+ (-0.159710408124)
- (-.0625)

Notice that the "imaginary" portions of p & q sum to zero and so we have:

X1 = 2*(.7923967592303) -0.159710408124 +.0625
X1 = 1.5847935184606 -0.097210408124
X1 = 1.48758311033


X2= p - q - r -s =
+ (0.7923967592303 + i* 0.6498360995438)
- (0.7923967592303 - i* 0.6498360995438)
- (-0.159710408124)
- (-.0625)

Here, the "real" portions of p & q sum to zero and so:
X2 = 2*(.6498360995438*i) + .159710408124 + .0625
X2 = 0.222210408124 + i*1.29967219908


X3= -p + q - r -s =
- (0.7923967592303 + i* 0.6498360995438)
+ (0.7923967592303 - i* 0.6498360995438)
- (-0.159710408124)
- (-.0625)

Again, the "real" portions of p & q sum to zero and so:
X3 = -2*(.6498360995438*i) + .159710408124 + .0625
X3 = 0.222210408124 - i*1.29967219908


X4= -p - q + r -s =
- (0.7923967592303 + i* 0.6498360995438)
- (0.7923967592303 - i* 0.6498360995438)
+ (-0.159710408124)
- (-.0625)

Here, the "imaginary" portions of p & q sum to zero and so:

X4 = -2*(.7923967592303)   -0.159710408124 +.0625
X4 = -1.5847935184606   -0.097210408124
X4 = -1.68200392658


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