To solve the quadratic equation 6x^2 - x - 12 = 0, we can use factoring, completing the square, or the quadratic formula.

Factoring:
We need to find two numbers whose product is -72 (the product of the coefficient of x^2 and the constant term) and whose sum is -1 (the coefficient of x). The numbers are -4 and 18. So, we can rewrite the equation as:
(6x + 3)(x - 4) = 0
Setting each factor equal to zero, we have:
6x + 3 = 0 or x - 4 = 0
Solving each equation, we get:
6x = -3 or x = 4
x = -3/6 or x = 4
x = -1/2 or x = 4

Completing the square:
We can rewrite the equation as:
6(x^2 - (1/6)x) = 12
To complete the square, we need to add and subtract (1/12)^2 = 1/144 inside the parentheses:
6(x^2 - (1/6)x + 1/144 - 1/144) = 12
Simplifying, we have:
6((x - 1/12)^2 - 1/144) = 12
Dividing both sides by 6, we get:
(x - 1/12)^2 - 1/144 = 2
(x - 1/12)^2 = 2 + 1/144
(x - 1/12)^2 = 289/144
Taking the square root of both sides, we have:
x - 1/12 = ±√(289/144)
x - 1/12 = ±(17/12)
x = 1/12 ± (17/12)
x = (1 ± 17)/12
x = 18/12 or x = -16/12
Simplifying, we get:
x = 3/2 or x = -4/3

Quadratic formula:
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the equation 6x^2 - x - 12 = 0, a = 6, b = -1, and c = -12. Substituting these values into the quadratic formula, we have:
x = (-(-1) ± √((-1)^2 - 4(6)(-12))) / (2(6))
x = (1 ± √(1 + 288)) / 12
x = (1 ± √289) / 12
x = (1 ± 17) / 12
x = 18/12 or x = -16/12
Simplifying, we get:
x = 3/2 or x = -4/3

Therefore, the solutions to the equation 6x^2 - x - 12 = 0 are x = -1/2, x = 4, x = 3/2, and x = -4/3.