Study sheet: Complex Quadratic Equations

📋 Course Outline

1. Square Roots of Complex Numbers
2. Worked Complex Root Examples
3. General Complex Quadratic Formula
4. Root Relations and Special Cases

📖 1. Square Roots of Complex Numbers

🔑 Key Concepts & Definitions

• Square root of a complex number : For $a=\alpha+i\beta$ and $z=x+iy$, the equation $z^2=a$ is equivalent to $x^2-y^2=\alpha$, $x^2+y^2=\sqrt{\alpha^2+\beta^2}$, and $2xy=\beta$.

📝 Essential Points

• To solve $z^2=a$, write $z=x+iy$, equate real and imaginary parts, add and subtract the resulting equations to find $x^2$ and $y^2$, then select sign pairs satisfying $2xy=\beta$.

💡 Memory Hook

Set z=x+iy, compare real and imaginary parts, then solve for x and y.

📖 2. Worked Complex Root Examples

📝 Essential Points

• The equation $z^2=3+4i$ has the two solutions $z=2+i$ and $z=-2-i$.

• For $z^2=-6-8i$, the equations give $x^2=2$ and $y^2=8$, and the two solutions are $z=\sqrt2-2\sqrt2i$ and $z=-\sqrt2+2\sqrt2i$.

💡 Memory Hook

Positive imaginary part gives x and y the same sign, whereas negative imaginary part gives opposite signs.

📖 3. General Complex Quadratic Formula

🔑 Key Concepts & Definitions

• Complex quadratic equation : an equation of the form $az^2+bz+c=0$, where $a\in\mathbb C^*$ and $$b,c\in\mathbb C$
• Discriminant : $b^2-4ac$, which is a complex number

📝 Essential Points

📐 Formula — If $\delta$ is a square root of the discriminant, then $\delta^2=\Delta$ and the two roots are $z'=\frac{-b+\delta}{2a}$ and $z''=\frac{-b-\delta}{2a}$.

💡 Memory Hook

Normalize, complete the square, define Δ, choose δ, then obtain the two roots.

📖 4. Root Relations and Special Cases

📝 Essential Points

📐 Formula — For the two roots of $az^2+bz+c=0$, the sum is $z'+z''=-\frac ba$ and the product is $z'z''=\frac ca$.

📌 If $a+b+c=0$, the roots are $1$ and $c/a$.

💡 Memory Hook

The discriminant gives the roots directly, whereas a±b+c=0 gives roots ±1 immediately.

📊 Synthesis Tables

Special coefficient conditions

Condition	Guaranteed root	Other root
a+b+c=0	1	c/a
a-b+c=0	−1	−c/a