Final answer:
The equation −6x²+24x+12=0 has a positive discriminant which is a perfect square, indicating that its roots are real, rational, and since the square root of the discriminant is not an integer, the roots are unequal.
Step-by-step explanation:
To determine if the roots of the equation −6x²+24x+12=0 are real, rational, imaginary, or irrational, we can calculate the discriminant from the quadratic formula. The discriminant is found by the formula b² - 4ac, where a, b, and c are coefficients of the equation ax² + bx + c = 0. In this case, a = -6, b = 24, and c = 12.
Lets calculate the discriminant:
discriminant = b² - 4ac = (24)² - 4(-6)(12) = 576 + 288 = 864
The discriminant is positive, which means the equation has two real roots. Since it is a perfect square (144²), the roots are also rational. Finally, as the discriminant does not yield a perfect square of an integer under the square root, we can deduce that the equation has two real, rational, and unequal roots.
Therefore, the correct answer is: d. real, rational, and unequal.