Solving the Quadratic Equation: 5x + 6x² + 3 = 0
To solve the quadratic equation 5x + 6x² + 3 = 0, we first need to rearrange it in the standard form:
6x² + 5x + 3 = 0
The quadratic formula is given by:
x = (-b ± √(b² – 4ac)) / 2a
In this formula:
- a is the coefficient of x² (which is 6),
- b is the coefficient of x (which is 5),
- c is the constant term (which is 3).
Now, we will first calculate the discriminant (b² – 4ac):
b² – 4ac = 5² – 4(6)(3)
= 25 – 72 = -47
Since the discriminant is negative, it indicates that there are no real solutions to this quadratic equation; instead, we will have complex solutions.
Now, let’s substitute the values of a, b, and c into the quadratic formula:
x = (-5 ± √(-47)) / (2 * 6)
We can simplify this as:
x = (-5 ± i√47) / 12
Thus, the two complex solutions for x can be expressed as:
- x₁ = (-5 + i√47) / 12
- x₂ = (-5 – i√47) / 12
In conclusion, the solutions to the equation 5x + 6x² + 3 = 0 are:
x₁ = (-5 + i√47) / 12 and x₂ = (-5 – i√47) / 12