Final answer:
To find the smallest integer value of k that makes the solutions of the quadratic equation 12x^2 + 11x + k = 0 complex, we need to look at the discriminant of the equation.
Step-by-step explanation:
To find the smallest integer value of k that makes the solutions of the quadratic equation 12x^2 + 11x + k = 0 complex, we need to look at the discriminant of the equation. The discriminant is given by b^2 - 4ac, where a, b, and c are the coefficients of the equation.
In this case, a = 12, b = 11, and c = k. For complex solutions, the discriminant must be negative. So we have:
11^2 - 4(12)(k) < 0
121 - 48k < 0
48k > 121
k > 2.52
The smallest integer value of k that makes the solutions complex is 3.