Final answer:
The solutions to the equation x²+9x+12=6 are found using the quadratic formula. After rearranging the equation to standard form and applying the formula, the solutions are x = -9/2 + √(57)/2 and x = -9/2 - √(57)/2, which corresponds to choice D.
Step-by-step explanation:
To find the solutions to the quadratic equation x²+9x+12=6, we first need to rewrite it in the standard form ax²+bx+c=0 by subtracting 6 from both sides, which gives us x²+9x+6=0. Next, we can apply the quadratic formula, which for a quadratic equation in the form ax²+bx+c=0 is given by x = −b ± √(b² − 4ac) / (2a), where a, b, and c are coefficients from the quadratic equation.
Using the quadratic formula with our coefficients a=1, b=9, and c=6, we find the discriminant first: b²-4ac = 9²-4×1×6 = 81-24 = 57. Substituting these values into the formula, we get:
x = −(9) ± √(57) / (2×1)
Thus, simplifying further, we have:
x = − 9/2 ± √(57)/2
So the solutions to the equation are x=- 9/2 + √(57)/2 and x=- 9/2 - √(57)/2, which corresponds to option D from the provided choices.