To find the solutions of the equation x² + 5x + 8 = 0, we can utilize the quadratic formula. The quadratic formula is given by:
x = (-b ± √(b² – 4ac)) / 2a
For our equation, the coefficients are:
- a = 1
- b = 5
- c = 8
Now we can plug these values into the quadratic formula:
x = (-5 ± √(5² – 4 * 1 * 8)) / (2 * 1)
Calculating the discriminant:
Discriminant = b² – 4ac = 5² – 4 * 1 * 8 = 25 – 32 = -7
Since the discriminant is negative (-7), this tells us that the solutions are complex (or imaginary) numbers.
Now, substituting the values back into the formula, we get:
x = (-5 ± √(-7)) / 2
We can express the square root of -7 using the imaginary unit i, where i = √(-1). Thus:
√(-7) = i√7
Now we rewrite our solutions:
x = (-5 ± i√7) / 2
So, the solutions to the equation x² + 5x + 8 = 0 are:
x = (-5 + i√7) / 2 and x = (-5 – i√7) / 2
In conclusion, the equation doesn’t have real solutions, but it has two complex solutions:
x = -2.5 + (√7/2)i and x = -2.5 – (√7/2)i.