If the equation of a parabola has a root of x = 2 + 5i, what is the other root and the equation of the parabola.

The equation of the parabola has a complex root x = 2 +
5i. Complex roots always come in pairs of the form a + b*i and a -
b*i.

As the parabola here has a root 2 + 5i, the other root
is the complex conjugate which is 2 - 5i.

To find the
equation of the parabola, we have to expand (x - (2 + 5i))((x - (2 - 5i)) =
y

(x - (2 + 5i))((x - (2 - 5i)) =
y

=> (x - 2 - 5i)(x - 2 + 5i) =
y

=> (x - 2)^2 - (5i)^2 =
y

=> x^2 + 4 - 4x - 5*i^2 =
y

=> x^2 + 4 - 4x + 5 =
y

=> y = x^2 - 4x +
9

The required root of the parabola is 2 - 5i
and the equation of the parabola is y = x^2 - 4x +
9