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

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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:

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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

Example # 3

Quartic Equation With No Real and 4 Complex Roots
2X4   + 4X3   + 6X2   + 8X + 10 = 0
Simplify the equation by dividing all terms by ‘a’, so the equation then becomes:

X4   +   2X 3   +   3X2  +   4X   +   5   =   0
Where   a  =  1   b  =  2   c  =  3   d   =   4   and   e   =   5
f = c – (3b2/8)   = 3 – (3*22/8)
f = 1.5
g = d + (b3 / 8) – (b*c/2)   =   4 + (23 / 8) – ((2 * 3) / 2) =
g = 2
h = e – (3*b4/256) + (b2 * c/16) – ( b*d/4)   =   5 – (3*24/256) + (22 * 3/16) – ( 2*4/4)
h = 3.5625
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 +.75*Y2 -0.75*Y -0.0625 = 0
Next, we obtain the 3 roots of this cubic equation by going to the:

CUBIC   EQUATION   CALCULATOR.
and the roots are:

Y1=  0.62065322983065
Y2=  -1.29275744640077
Y3=  -0.07789578342988
Let ‘p’ and ‘q’ be the square roots of ANY 2 non-zero roots (Y1 Y2 or Y3).
In this case, let us take the square roots of the 2 negative numbers.

To find the square root of a negative number:
• take the square root of the absoute value of the number then
• multiply it by “i”.
For example, the square root of -9 is 3i.

p=SqRoot(Y2) = 1.13699491925020i
q=SqRoot(Y3) = 0.279098160921709i
r= -g/(8*pq)   = -2/(8*-0.317333190940058)   = 0.787815479557647
s= b/(4*a) = 2/(4*1) = .5
Then the four roots of the quartic equation are:

X1= p + q + r -s =
1.1369949192502i + 0.279098160921709i + 0.787815479557647 -.5 =
+1.41609308017191i   +0.28781547955765
X1 = 0.28781547955765   + i*1.41609308017191

X2= p – q – r -s =
1.1369949192502i  -0.279098160921709i  -0.787815479557647  -.5
0.857896758328492i   -1.287815479557650
X2 = -1.287815479557650   + i*0.857896758328492

X3= -p + q – r -s =
-1.1369949192502i   +0.279098160921709i   -0.787815479557647   -.5
-0.857896758328492i   -1.28781547955765
X3 = -1.28781547955765   – i*0.857896758328492

X4= -p – q + r -s =
-1.1369949192502i   -0.279098160921709i   +0.787815479557647   -.5
-1.41609308017191i   +0.287815479557647
X4 = 0.287815479557647   – i*1.41609308017191

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