Final answer:
d. Real, rational, and unequal. The roots of the equation -6x^2 + 24x + 12 = 0 are real, rational, and unequal.
Step-by-step explanation:
The roots of the equation -6x^2 + 24x + 12 = 0 are:
a. Real, rational, and equal.
b. Imaginary.
c. Real, irrational, and unequal.
d. Real, rational, and unequal.
To find the roots of this quadratic equation, we can use the quadratic formula. For an equation of the form ax^2 + bx + c = 0, the roots can be calculated using the formula:
x = (-b +- sqrt(b^2 - 4ac)) / (2a)
In this equation, a = -6, b = 24, and c = 12. Plugging these values into the formula, we get:
x = (-24 +- sqrt(24^2 - 4*(-6)*12)) / (2*(-6))
Simplifying the equation, we get:
x = (-24 +- sqrt(576 - (-288))) / (-12)
x = (-24 +- sqrt(864)) / (-12)
x = (-24 +- 29.39) / (-12)
So, the roots of the equation are real, rational, and unequal.