The roots of the quadratic function can be calculated using the quadratic formula:
Quadratic formula: q = (-b ± √(b² – 4ac)) / (2a)
In the given function, f(q) = q² + 125, we can identify the coefficients as follows:
- a = 1 (the coefficient of q²)
- b = 0 (there’s no q term)
- c = 125 (the constant term)
Substituting these values into the quadratic formula:
1. Calculate the discriminant: b² – 4ac = 0² – 4 * 1 * 125 = 0 – 500 = -500
2. Since the discriminant is negative, there are no real roots. Instead, we will have complex roots.
3. Using the quadratic formula:
- q = (-0 ± √(-500)) / (2 * 1)
- q = 0 ± √(500)i / 2
- √(500) can be simplified to 10√5, so:
- q = 0 ± (10√5)i / 2
- q = 5√5i and q = -5√5i
In conclusion, the roots of the quadratic function f(q) = q² + 125 are:
- q = 5√5i
- q = -5√5i
These roots are complex numbers and indicate that the parabola described by the quadratic function does not intersect the x-axis.