##### Quartic Equation Definition:

In algebra, a quartic function is defined as a function of the form **ax**^{4}+ bx^{3} + cx^{2} + dx + e = 0, where 'a' is non zero, which is defined by a fourth degree polynomial, called a quartic polynomial.

#### Quartic Equation Formula:

ax^{4} + bx^{3}+ cx^{2} + dx + e = 0
###### where,

a = coefficient of x^{4}
b = coefficient of x^{3}
c = coefficient of x^{2} and
d = coefficient of x.
e = constant.
#### Quartic Equation solving formula:

x_{1} = p + q + r - s
x_{2} = p - q - r - s
x_{3} = -p + q - r - s
x_{4} = -p - q + r - s
##### Example 1:

Calculate the roots (x1, x2, x3, x4) of the quartic equation,
3X^{4} + 6X^{3} - 123X^{2} - 126X + 1080 = 0

###### Step 1:

From the above equation, the value of a = 3, b = 6, c = -123, d = -126, e = 1080.

###### Step 2:

To find x :
Substitute the values in the formulas below.
f = c - ( 3b ² / 8 )
g = d + ( b ³ / 8 ) - ( b x c / 2 )
h = e - ( 3 x b^{4} / 256 ) + ( b ² x c / 16 ) - ( b x d / 4 )

###### Step 3:

Form as Cubic Equation :
y ³ + ( f / 2 ) y ² + (( f ² - 4 x h ) / 16 ) y - g ² / 64 = 0
where,
a = coefficient of y ³
b = coefficient of y²
c = coefficient of y
d = constant

###### Step 4:

From the above equation, the values of a, b, c, and d are,
a = 1,
b = f / 2,
c = (( f ² - 4 x h ) / 16 ), and
d = - g² / 64.

###### Step 5:

To Find y:
Substitute the values in the formula's below to find the roots. The variable disc is nothing but the discriminant, denoted generally as delta(Δ)
**discriminant(Δ) = q**^{3} + r^{2}
q = (3c - b^{2}) / 9
r = -27d + b(9c - 2b^{2})
s = r +√ (discriminant)
t = r - √(discriminant)
term1 = √(3.0) * ((-t + s) / 2)
r_{13} = 2 * √(q)
y_{1} = (- term1 + r_{13}*cos(q^{3}/3) )
y_{2} = (- term1 + r_{13}*cos(q^{3}+(2∏)/3) )
y_{3} = (- term1 + r_{13}*cos(q^{3}+(4∏)/3) )

###### Step 6:

We get the roots,
y_{1} = 20.25 , y_{2} = 0 and y_{3} = 1.

###### Step 7:

After finding cubic equation solve quartic equation
Substitue y_{1}, y_{2}, y_{3} in p, q, r, s.
NOTE : Let p and q be the square root of any 2 non-zero roots.
p = sqrt(y1) = 4.5
q = sqrt(y3) = 1
r = -g / (8pq) = 0
s = b / (4a) = 0.5

###### Step 8:

We get the roots, **x1 = 5, x2 = 3, x3 = -4 and x4 = -6.**