Solving Cubic Equations With Worked Examples

Cubic equation is a third degree polynomial equation. To solve this equation, write down the formula for its roots, the formula should be an expression built with the coefficients a, b, c and fixed real numbers using only addition, subtraction, multiplication, division and the extraction of roots. Here given are worked examples for solving cubic equations.

Example 1:

Let us consider the problem with a cubic equation 5x3 + 4x2+ 2x + 2

Solution:

We can calculate the value using the given formula.

Cubic Equation Formula:

x1=(- term1 + r13 x cos(q3/3) ) x2=(- term1 + r13 x cos(q3+(2 x Π)/3) ) x3=(- term1 + r13 x cos(q3+(4 x Π)/3) ) Where,

discriminant(Δ) = q3 + r2 term1 = √(3.0) x ((-t + s)/2) r13= 2 x √(q) q = (3c- b2)/9 r = -27d + b(9c-2b2) s = r + √(discriminant) t = r - √(discriminant)

Substituting the values in the formula,

discriminant(Δ) = q3 + r2

= [(3x2 - 42) / 9]3 + [-27x2 + 3(9x2 - 2x32)]2

= [(6-16)/9]3 + [-54 + 3(18 - 62)]2

= [-10 / 9]3 + [-54 + 3(18 - 36)]2

= [-1.11111]3 + [-54 + 3(-18)]2

= -1.3717 + [-54 - 54]2

= -1.3717 + -11664

(Δ) = -11665.3717

Find the value of s,

s = r + √(discriminant)

= -108 + √-11665.3717

Hence, s = -108 + 108.0063502762684783i

Find the value of t,

t = r - √(discriminant)

= -108 - √-11665.3717

Hence, t = -108 - 108.0063502762684783i

Find the value of term1,

term1 = √(3.0) x ((-t + s)/2)

= √(3.0) x ((-(-108 - 108.0063502762684783i) + (-108 + 108.0063502762684783i)/2)

= 1.732 x ((108 + 108.0063502762684783i - 108 - 108.0063502762684783i) / 2)

Hence, term1 = 1.732 x 0 = 0