Complex Numbers and Quadratic Equations

Formulas for Complex Numbers and Quadratic Equations

Solutions to a quadratic equation

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ For ax^2 + bx + c = 0. If b^2 - 4ac < 0, the roots are complex.

Modulus of a complex number

$|z| = \sqrt{a^2 + b^2}$ For a complex number z = a + ib.

Polar and Euler's form

$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$ r is the modulus |z| and θ is the argument arg(z).

Multiplication in polar form

$z_1 z_2 = r_1 r_2 (\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2))$ Moduli multiply and arguments add.

Division in polar form

$\frac{z_1}{z_2} = \frac{r_1}{r_2} (\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2))$ Moduli divide and arguments subtract.

De Moivre's theorem

$(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$ Holds for any integer n.

nth roots of a complex number

$z^{1/n} = r^{1/n} \left( \cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right) \right)$ k = 0, 1, 2, ..., n-1, giving n distinct roots.

Sum of nth roots of unity

$1 + \omega + \omega^2 + ... + \omega^{n-1} = 0$ Where ω is a non-real nth root of unity.

Rotation formula

$\frac{z_3 - z_1}{z_2 - z_1} = \frac{|z_3 - z_1|}{|z_2 - z_1|} e^{i\alpha}$ Vector z_1z_2 is rotated by angle α to become vector z_1z_3.