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Quartic Equation Tutorial

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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 - ( 3b ² / 8 )
    g = d + ( b ³ / 8 ) - ( b x c / 2 )
    h = e - ( 3 x b4 / 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 y ²
        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 = - g ² / 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 +√ (discriminant)
     t = r - √(discriminant)
     term1 = √(3.0) * ((-t + s) / 2)
     r13 = 2 * √(q)
     y1=(- term1 + r13*cos(q3/3) )
     y2=(- term1 + r13*cos(q3+(2∏)/3) )
     y3=(- term1 + r13*cos(q3+(4∏)/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.




This tutorial will help you dynamically to find the roots of a Quartic Equation.