To find the zeros of the function f(x) = x² + 2x + 3, we need to determine the values of x that make the equation equal to zero. In mathematical terms, we set the function equal to zero:
x² + 2x + 3 = 0
This is a quadratic equation, and we can find the zeros using the quadratic formula:
x = (-b ± √(b² – 4ac)) / 2a
where a = 1, b = 2, and c = 3. Now, let’s plug in the values:
b² – 4ac = 2² – 4(1)(3) = 4 – 12 = -8
Since the discriminant (b² – 4ac) is negative, this means there are no real solutions for the equation (no real zeros). However, we can find the complex zeros. We continue with the quadratic formula:
x = (-2 ± √(-8)) / 2(1)
This simplifies to:
x = (-2 ± 2i√2) / 2
Breaking it down further, we get:
x = -1 ± i√2
Thus, the zeros of the function f(x) = x² + 2x + 3 are:
x = -1 + i√2 and x = -1 – i√2
In summary, the function has two complex zeros, indicating it does not cross the x-axis in the Cartesian plane.