A quadratic equation has the format ax^2 + bx + c = 0, where a, b, and c are constants. The roots, or solutions, of a quadratic equation can be complex numbers which are of the form a + bi, where i squared is -1 and therefore we say i is an imaginary number.
The roots of a quadratic equation always come in pairs. If given that -87i is a root of the equation f(X) = X^2 + 7569 = 0, we know that this is a quadratic equation where a = 1, b = 0 and c = 7569, and the roots are x1 = 87i and x2 = -87i. This is due to the fact that in case of complex roots, they are always in a conjugate pair. The conjugate of a complex number is obtained by changing the sign of its imaginary part.
So, given that one root of the equation is -87i, the other root would then be its conjugate which is 87i.
Therefore, the other root of the given function f(x) = x^2 + 7569 = 0 is 87i.