aх^3 + bх^2 + cx + d = 0 is a generalized form of a cubic equation.

The conventional method for solving a cubic equation is to convert it to a quadratic equation and then solve it using factoring or the quadratic formula. A cubic equation always has at least one actual root, unlike a quadratic equation, which may have no genuine solution.

Determine the roots of the cubic equation

2х^3 + 3х^2 – 11x – 6 = 0

The possible factors are 1, 2, 3, and 6 (since d is 6).

Here, you need to apply the Factor Theorem to check the possible values

f (1) = 2 + 3 – 11 – 6 ≠ 0

f (–1) = –2 + 3 + 11 – 6 ≠ 0

f (2) = 16 + 12 – 22 – 6 = 0

So, x = 2 is the first root.

We can get the other roots of the equation using the synthetic division method.

= (x – 2) (ax2 + bx + c)

= (x – 2) (2x2 + bx + 3)

= (x – 2) (2x2 + 7x + 3)

= (x – 2) (2x + 1) (x +3)

Answer: x = 2, x = -(1/2), and x = -3

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