To solve the quadratic equation 6x² + 5x + 3 = 0 using the quadratic formula, we start by identifying the coefficients:
- a = 6
- b = 5
- c = 3
The quadratic formula is given by:
x = (-b ± √(b² – 4ac)) / (2a)
First, we need to calculate the discriminant (b² – 4ac):
b² = 5² = 25
4ac = 4 * 6 * 3 = 72
Now, compute the discriminant:
Discriminant = 25 – 72 = -47
Since the discriminant is negative, this means there are no real solutions; instead, we will have complex solutions.
Now we plug the values into the quadratic formula:
x = (–5 ± √(-47)) / (2 * 6)
This simplifies to:
x = (–5 ± i√47) / 12
Thus, the two complex solutions for x are:
- x₁ = (–5 + i√47) / 12
- x₂ = (–5 – i√47) / 12
In conclusion, the values of x for the equation 6x² + 5x + 3 = 0 are complex numbers due to the negative discriminant.