Equations with Complex Roots — practice questions

Solve $x^2 + 2x + 10 = 0$, giving the roots in the form $a + bi$.

A quadratic equation with real coefficients has $-2 + 5i$ as one of its roots and a leading coefficient of 3. Find the equation in the form $ax^2 + bx + c = 0$.

Prove that the equation $x^2 + kx + (k + 3) = 0$ has real roots for no values of $k$ in the interval $-2 < k < 6$, and determine the exact values of $k$ at which the nature of the roots changes.

Show that $(z - 2)$ is a factor of $P(z) = z^3 - 2z^2 + 9z - 18$, and find the remaining roots.

The cubic $z^3 + 2z^2 + az + b = 0$ has real coefficients and one root $z = -1 + 2i$. Find $a$, $b$, and the real root.

A cubic equation with real coefficients has a root $z = 3 + i$ and satisfies $P(0) = -30$. Given that the cubic is $P(z) = a(z^3 + bz^2 + cz + d)$ with $a$, $b$, $c$, $d$ real and the leading coefficient $a$ equal to 1, find the equation and all three roots. Justify why the third root must be real.