Final answer:
The quadratic equation 3x²+12x+9=0 has two real solutions, which can be found using the quadratic formula. After calculation, the solutions are x = -1 and x = -3.
Step-by-step explanation:
To find the real solutions to the quadratic equation 3x²+12x+9=0, we can use the quadratic formula, which is applicable to any equation of the form ax²+bx+c=0. The quadratic formula is:
x = ∛((-b ± √(b²-4ac))/(2a))
In our equation, a = 3, b = 12, and c = 9. Plugging these values into the quadratic formula:
x = ∛((-12 ± √(12²-4(3)(9)))/(2(3)))
Simplifying inside the square root:
x = ∛((-12 ± √(144-108))/(6))
Further simplifying:
x = ∛((-12 ± √36)/(6))
Since √36 is 6, the equation then becomes:
x = (-12 ± 6)/6
There are two solutions:
- x = (-12 + 6)/6 = -1
- x = (-12 - 6)/6 = -3
Therefore, the quadratic equation 3x²+12x+9=0 has two real solutions: x = -1 and x = -3.