Complex Numbers & Quadratic Equations Cheatsheet

Cheat Codes & Shortcuts

• Complex Number: \( z = a + bi \) where \( i^2 = -1 \)
• Modulus: \( |z| = \sqrt{a^2 + b^2} \)
• Conjugate: \( \overline{z} = a - bi \)
• Polar Form: \( z = r(\cos \theta + i \sin \theta) \), where \( r = |z| \) and \( \theta = \arg(z) \)
• Euler's Formula: \( z = r e^{i \theta} \)
• Quadratic Equation Standard Form: \( ax^2 + bx + c = 0 \)
• Discriminant: \( \Delta = b^2 - 4ac \)
• Roots of Quadratic: \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \)
• Nature of Roots: Real and distinct if \( \Delta > 0 \), real and equal if \( \Delta = 0 \), complex if \( \Delta < 0 \)
• Sum and Product of Roots: \( \alpha + \beta = -\frac{b}{a} \), \( \alpha \beta = \frac{c}{a} \)

Quick Reference Table

Type	Form	Solution / Property
Complex Number	\( z = 3 + 4i \)	\( |z| = 5 \), \( \overline{z} = 3 - 4i \)
Polar Form	\( z = 1 + i \)	\( r = \sqrt{2} \), \( \theta = \frac{\pi}{4} \), so \( z = \sqrt{2} e^{i\pi/4} \)
Quadratic Equation	\( 2x^2 - 4x + 2 = 0 \)	\( \Delta = 0 \Rightarrow \) One real root \( x = 1 \)
Quadratic Roots (Complex)	\( x^2 + 4x + 5 = 0 \)	\( \Delta = -4 \Rightarrow x = -2 \pm i \)
Sum & Product of Roots	\( ax^2 + bx + c = 0 \)	\( \sum = -\frac{b}{a}, \quad \prod = \frac{c}{a} \)

Advice

Complex Numbers & Quadratic Equations Quick Tips

• Modulus & Argument: Find magnitude and angle to express complex numbers in polar form.
• Euler's Formula: Useful for powers and roots of complex numbers.
• Quadratic Roots: Use quadratic formula, consider discriminant carefully.
• Nature of Roots: Distinguish real vs complex from discriminant.
• Sum & Product: Use relationships without solving quadratic.

Complex Numbers & Quadratic Equations Speed Quiz

Test your speed with 5 questions! You have 30 seconds per question.