The quadratic formula is used to find the solutions (or roots) of a quadratic equation of the form ax² + bx + c = 0. In this case, the given equation is 2x² + 5x + 5 = 0.
First, identify the coefficients:
- a = 2
- b = 5
- c = 5
Next, plug these values into the quadratic formula, which is:
x = (-b ± √(b² – 4ac)) / (2a)
Now, calculate the discriminant (b² – 4ac):
- Calculate b²:
5² = 25 - Calculate 4ac:
4 × 2 × 5 = 40 - Now, find b² – 4ac:
25 – 40 = -15
Since the discriminant is negative (-15), this means there are no real solutions; however, there are two complex solutions.
Now substitute back into the quadratic formula:
x = (-5 ± √(-15)) / (2 × 2)
This simplifies to:
x = (-5 ± i√15) / 4
The final complex solutions can be expressed as:
- x = -5/4 + i√15/4
- x = -5/4 – i√15/4
Thus, the solutions for the equation 2x² + 5x + 5 = 0 are:
- x = -5/4 + i√15/4
- x = -5/4 – i√15/4