The roots of the quadratic equation 3x² + 4x + 5 = 0 can be found using the quadratic formula:
x = (-b ± √(b² – 4ac)) / 2a
In this equation, a = 3, b = 4, and c = 5. First, let’s calculate the discriminant, which is the part under the square root:
Discriminant = b² – 4ac
When we substitute the values:
Discriminant = 4² – 4 × 3 × 5
Discriminant = 16 – 60 = -44
Since the discriminant is negative (-44), this indicates that the quadratic equation does not have any real roots; instead, it has two complex roots.
Next, we can express the roots using the quadratic formula:
x = (-4 ± √(-44)) / (2 × 3)
This simplifies to:
x = (-4 ± 2i√11) / 6
Finally, we can break this down further to find:
x = -2/3 ± (i√11)/3
Thus, the roots of the equation 3x² + 4x + 5 = 0 are:
- x = -2/3 + (i√11)/3 (a complex root)
- x = -2/3 – (i√11)/3 (a complex root)
In summary, the equation has two complex roots, each expressed as a combination of real and imaginary parts, reflecting that for this particular quadratic, real intersections with the x-axis do not exist.