Answer to Question #114747 in Algebra for ANJU JAYACHANDRAN

Question

Find the roots of 2x^4+9x^3-9x^2-46x+24 by Ferrari’s method.

Answer 1

Let 2x⁴ + 9x³ - 9x² - 46x + 24 = 0

Dividing the whole equation by 2, we get,

x⁴ + (9/2)x³ - (9/2)x² - 23x + 12 = 0

Rewriting the above equation as :

x⁴ + (9/2)x³ = (9/2)x² + 23x - 12

Adding (81/16)x² to both sides of the above equation :

x⁴ + (9/2)x³ + (81/16)x² = (9/2)x² + 23x - 12 + (81/16)x²

i.e., (x² + (9/4)x)² = (9/2 + 81/16)x² + 23x - 12

(x² + (9/4)x)² = (153/16)x² + 23x - 12

Let y be another variable, consider the equation :

(x² + (9/4)x + (y/2))² = (153/16)x² + 23x - 12 +y(x² + (9/4)x) + y²/4

i.e., (x² + (9/4)x + (y/2))² = ((153/16) + y)x² + (23 + (9y/4))x + ((y²/4) - 12) ... eq. (1)

The RHS of equation 1 is a square of linear polynomial if and only if b² - 4ac = 0.

Here, a = (153/16) + y

b = 23 + (9y/4)

c = (y²/4) - 12

Calculating b² - 4ac :

(23 + (9y/4))² - 4 × ((153/16) + y) × ((y²/4) - 12) = 0

(81/16)y² + (207/2)y + 529 - 4 × ((153/64)y² - (459/4) + (y³/4) - 12y) = 0

(81/16)y² + (207/2)y + 529 - (153/16)y² + 459 - y³ + 48y = 0

y³ + (72/16)y² - (303/2)y - 988 = 0

i.e., y³ + (9/2)y² - (303/2)y - 988 = 0

Multiplying the above equation by 2, we get :

2y³ + 9y² - 303y - 1976 = 0

Solving the above cubic equation, we get the following roots for y as :

y = 13, y = -8, y = -(19/2)

Substituting y = -(19/2) in equation (1), we get :

(x² + (9/4)x - (19/4))² = ((153/16) - (19/2))x² + (23 + (171/8))x + ((361/16) - 12)

i.e., (x² + (9/4)x - (19/4))² = (1/16)x² + (13/8)x + (169/16)

i.e., (x² + (9/4)x - (19/4)) = ((1/4)x + (13/4))²

(x² + (9/4)x - (19/4)) - ((1/4)x + (13/4))² = 0

i.e., (x² + (9/4)x - (19/4) + (1/4)x + (13/4)) × (x² + (9/4)x - (19/4) - (1/4)x - (13/4)) = 0

(x² + (5/2)x - (3/2)) × (x² + 2x - 8) = 0

Therefore,

x² + (5/2)x - (3/2) = 0 & x² + 2x - 8 = 0

We get the roots for the above quadratic equations as :

x = (1/2), -3 and x = 2, -4 respectively.

Therefore, using the Ferrari's method, the roots for the given equation are :

x = (1/2), 2, -3, -4