To find the zeroes of the function f(x) = x^2 – 2x + 3, we need to solve the equation f(x) = 0. This means we set the equation equal to zero:
x^2 – 2x + 3 = 0
Next, we can use the quadratic formula to find the roots of this quadratic equation, which is:
x = -b ± √(b² – 4ac)}{2a}
In our equation, a = 1, b = -2, and c = 3. Plugging these values into the formula, we get:
x = 2 ± √((-2)² – 4(1)(3))}{2(1)}
This simplifies to:
x = 2 ± √(4 – 12)}{2}
x = 2 ± √(-8)}{2}
Since the discriminant (4 – 12 = -8) is negative, there are no real roots for this equation. Instead, we have complex roots:
√(-8) = √(8) * √(-1) = 2√2i
Thus, we can express the roots as:
x = 2 ± 2√2i}{2}
Finally, simplifying this gives us the two zeroes:
x = 1 ± √2i
In conclusion, the zeroes of the function f(x) = x^2 – 2x + 3 are:
1 + √2i and 1 – √2i (complex roots).