What is the discriminant of \(z^2+4z+5=0\)?

State the complex roots of \(z^2+1=0\).

Solve \(z^2-4=0\) over \(\mathbb C\).

Solve \(z^2+9=0\).

Solve \(z^2+2z+2=0\).

Solve \(z^2-6z+13=0\).

Solve \(2z^2+2z+5=0\).

Solve \(3z^2-12z+15=0\).

Solve \((z-1)^2+4=0\).

Solve \(z^2+(2-2i)z-4i=0\).

Solve \(z^2+4z+8=0\) and verify one root.

Explain why the roots of \(z^2-2z+10=0\) must be conjugates, then find them.

Find a monic quadratic with real coefficients whose roots are \(2+i\) and \(2-i\).

Find \(p\) and \(q\) if \(z^2+pz+q=0\) has roots \(-1+4i\) and \(-1-4i\).

For which real \(m\) does \(z^2+2z+m=0\) have a repeated complex root?

For which real \(m\) does \(z^2+4z+m=0\) have two non-real roots?

Find all real \(m\) such that \(z=1+i\) is a root of \(z^2+mz+2=0\).

A learner says \(z^2+2z+5=0\) has no roots because the discriminant is negative. Diagnose and solve correctly.

Show directly that if \(u+iv\) is a root of a real-coefficient quadratic, then \(u-iv\) is also a root.

Use completing the square to solve \(z^2-2iz-2=0\).