To solve the quadratic equation 3x² + 2x + 7 = 0, we can use the quadratic formula, which is given by:
x = (-b ± √(b² – 4ac)) / 2a
In our equation, a = 3, b = 2, and c = 7. First, we need to calculate the discriminant, which is b² – 4ac:
Discriminant = 2² – 4(3)(7) = 4 – 84 = -80
Since the discriminant is negative (-80), this means that the quadratic equation has no real solutions, only complex (imaginary) solutions. We can proceed to find the complex solutions using the quadratic formula.
Now, substituting values into the quadratic formula:
x = (-2 ± √(-80)) / (2 * 3)
We simplify the square root of -80:
√(-80) = √(80) * √(-1) = √(16 * 5) * i = 4√5 * i
Now substitute it back:
x = (-2 ± 4√5 * i) / 6
Now we can divide each term:
x = -2/6 ± (4√5 * i)/6
x = -1/3 ± (2√5 * i) / 3
So, the solutions to the equation 3x² + 2x + 7 = 0 rounded to the nearest hundredth are:
x = -0.33 ± 1.49i
Therefore, the two complex solutions are approximately:
x₁ = -0.33 + 1.49i and x₂ = -0.33 – 1.49i.