To find the zeroes of the function f(x) = x² + 2x + 3, we need to solve the equation f(x) = 0. So we set up the equation:
x² + 2x + 3 = 0
This is a quadratic equation, and we can solve it using the quadratic formula, which is given by:
x = -b ± √(b² – 4ac) / 2a
In our equation, the coefficients are:
- a = 1
- b = 2
- c = 3
Now we can plug these values into the quadratic formula. First, we calculate the discriminant, b² – 4ac:
Discriminant = (2)² – 4(1)(3) = 4 – 12 = -8
Since the discriminant is negative, this means that there are no real solutions to the equation x² + 2x + 3 = 0. Instead, we have two complex solutions. We continue using the quadratic formula:
x = (-2 ± √(-8)) / (2 * 1)
We can simplify this further:
x = (-2 ± √(8)i) / 2
x = (-2 ± 2√2i) / 2
x = -1 ± √2i
So, the zeroes of the function f(x) = x² + 2x + 3 are:
x = -1 + √2i and x = -1 – √2i.