To solve the quadratic equation 2x² + 5x + 4 = 0 using the quadratic formula, we first need to identify the coefficients in the standard form of a quadratic equation, which is ax² + bx + c = 0.
In this case, we have:
- a = 2
- b = 5
- c = 4
The quadratic formula is given by:
x = (-b ± √(b² – 4ac)) / 2a
Next, we calculate the discriminant (b² – 4ac):
- b² = 5² = 25
- 4ac = 4 * 2 * 4 = 32
- So, the discriminant = 25 – 32 = -7
Since the discriminant is negative (-7), this means there are no real solutions; instead, there are two complex (imaginary) solutions. We can find these by continuing with the quadratic formula:
Plugging in the values:
x = (-5 ± √(-7)) / (2 * 2)
This can be simplified to:
x = (-5 ± i√7) / 4
Thus, the two complex solutions are:
x = (-5 + i√7) / 4 and x = (-5 – i√7) / 4
In summary, the solutions to the equation 2x² + 5x + 4 = 0 using the quadratic formula are:
- x = (-5 + i√7) / 4
- x = (-5 – i√7) / 4