Quartic Equation Definition:
A general quartic equation (also called a biquadratic equation) is a fourth-order polynomial equation of the form,
ax4 + bx3 + cx2 + dx + e = 0.
Quartic Equation Formula:
ax4 + bx3 + cx2 + dx + e = 0
where,
a = coefficient of x4
b = coefficient of x3
c = coefficient of x2 and
d = coefficient of x.
e = constant.
Quartic Equation solving formula:
x1 = p + q + r - s
x2 = p - q - r - s
x3 = -p + q - r - s
x4 = -p - q + r - s
Example 1: Calculate the roots(x1, x2, x3, x4) of the quartic equation,
3X4 + 6X3 - 123X2 - 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 - ( 3b² / 8 )
g = d + ( b³ / 8 ) - ( b x c / 2 )
h = e - ( 3 x b4 / 256 ) + ( b² x c / 16 ) - ( b x d / 4 )
Step 3: Form as Cubic Equation :
y³ + ( f / 2 ) y² + (( f ² - 4 x h ) / 16 ) y - g² / 64 = 0
where,
a = coefficient of y³
b = coefficient of y²
c = coefficient of y
d = constant
Step 4: From the above equation, the value of 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) = q3 + r2
q = (3c - b2) / 9
r = -27d + b(9c - 2b2)
s = r + math.sqrt(discriminant)
t = r - math.sqrt(discriminant)
term1 = math.sqrt(3.0) * ((-t + s) / 2)
r13 = 2 * math.sqrt(q)
y1=(- term1 + r13*math.cos(q3/3) )
y2=(- term1 + r13*math.cos(q3+(2*math.PI)/3) )
y3=(- term1 + r13*math.cos(q3+(4*math.PI)/3) )
Step 6: We get the roots,
y1 = 20.25 , y2 = 0 and y3 = 1.
Step 7: After finding cubic equation solve quartic equation
Substitue y1, y2, y3 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.
This is an example to calculate quartic equation.
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