Flashcards: Complex Quadratic Equations — 11 cards

What equations correspond to the square root of a complex number $a=\alpha+i\beta$?

They are $x^2 - y^2 = \alpha$, $x^2 + y^2 = \sqrt{\alpha^2 + \beta^2}$, and $2xy = \beta$.

Square root of complex number: a=α+iβ

Solve for z=x+iy: equations for x, y from complex parts.

What is the first step to solve $z^2 = a$ for complex $z$?

Write $z$ as $x + iy$ and equate real and imaginary parts.

Solutions to z²=3+4i

z=2+i and z=-2-i.

What are the two solutions of the equation $z^2=3+4i$?

The solutions are $z=2+i$ and $z=-2-i$.

Complex quadratic formula element

Discriminant is b²-4ac, complex number.

What are the values of $x^2$ and $y^2$ for $z^2=-6-8i$?

They are $x^2=2$ and $y^2=8$ respectively.

Roots relation: sum

Sum is -b/a.

Roots relation: product

Product is c/a.

Special root condition: a+b+c=0

Roots are 1 and c/a.

Root calculation memory hook

Normalize, complete square, find Δ, then roots.