To solve the quadratic equation 2x² + 5x + 5 = 0 using the quadratic formula, we need to identify the coefficients a, b, and c from the general quadratic form ax² + bx + c = 0. In this case:
- a = 2
- b = 5
- c = 5
The quadratic formula is given by:
x = (-b ± √(b² – 4ac)) / (2a)
Now, we will calculate the discriminant, which is the part under the square root:
D = b² – 4ac
Substituting the values:
D = 5² – 4(2)(5) = 25 – 40 = -15
Since the discriminant is negative (-15), this means there are no real solutions; instead, we will have complex solutions.
Now, applying the quadratic formula:
x = (–5 ± √(–15)) / (2(2))
This simplifies to:
x = (–5 ± i√15) / 4
Thus, the final answers are:
x = (–5/4) + (i√15)/4 and x = (–5/4) – (i√15)/4
These represent the two complex solutions of the original quadratic equation.