To find the solutions of the quadratic equation x² + 2x + 2 = 0, we can use the quadratic formula, which is given by:
x = −b ± √(b² − 4ac) / 2a
In this formula:
- a is the coefficient of x²,
- b is the coefficient of x,
- c is the constant term.
For our equation, we have:
- a = 1
- b = 2
- c = 2
Next, we will calculate the discriminant:
Discriminant (D) = b² − 4ac
D = (2)² – 4(1)(2) = 4 – 8 = -4
Since the discriminant is negative (D = -4), this means that the quadratic equation does not have real solutions. Instead, it has two complex solutions. To find these solutions, we continue with the quadratic formula:
x = −2 ± √(−4) / 2(1)
We know that √(−1) is equal to i (the imaginary unit), so:
√(−4) = 2i
Substituting this back into our formula:
x = −2 ± 2i / 2
Now, we simplify:
x = −1 ± i
In conclusion, the solutions to the equation x² + 2x + 2 = 0 are:
- x = -1 + i
- x = -1 – i
These are the two complex solutions to the given quadratic equation.