To find the roots of the quadratic equation 3x² + 4x + 5 = 0, we can use the quadratic formula, which is:
x = (-b ± √(b² – 4ac)) / (2a)
In our equation, a = 3, b = 4, and c = 5. First, we will calculate the discriminant (b² – 4ac):
Discriminant = 4² – 4 * 3 * 5 = 16 – 60 = -44
The discriminant is negative (-44), which indicates that the equation does not have real roots. Instead, it has two complex roots. We can proceed to find these complex roots using the quadratic formula:
x = (-4 ± √(-44)) / (2 * 3)
We can express the square root of the negative discriminant as follows:
√(-44) = √(44) * i = 2√(11) * i
Now substituting this back into the formula:
x = (-4 ± 2√(11)i) / 6
Simplifying this gives:
x = -2/3 ± (√(11)/3)i
Therefore, the roots of the equation 3x² + 4x + 5 = 0 are:
x = -2/3 + (√(11)/3)i and x = -2/3 – (√(11)/3)i.
This shows the nature of the roots as complex conjugates, which is typical when the discriminant is negative.