Understanding Complex Numbers and Quadratic Equations

Understanding Imaginary Numbers

√-25 has no real solution because (-5)² = 25 and (5)² = 25, never -25.

Definition of the Imaginary Unit

√-1 = i (imaginary unit). Therefore, √-25 = √25 * √-1 = 5i.

In general: √-any = √any * i. Example: √-200 = i√200 = 10i√2.

Powers of i

• i¹ = i
• i² = -1
• i³ = -i
• i⁴ = 1

This pattern repeats every four powers. Example: i¹⁵ = i¹² * i³ = -i.

Complex Numbers

Complex numbers are a combination of real and imaginary numbers. Standard form: a + bi (Real part + Imaginary part).

Operations with Complex Numbers

• Absolute Value: |4+3i| = √4²+3² = √16+9 = √25 = 5
• Conjugate: The conjugate of (4+3i) is (4-3i).
• Addition: (4+3i) + (2-i) = 6+2i
• Subtraction: (4+3i) - (2-i) = 2+4i
• Multiplication: (4+3i)*(2-i) = 8-4i+6i-3i² = 8+2i+3 = 11+2i
• Difference of Squares: (a+bi)(a-bi) = a²+b²

Quadratic Equations

Standard form: ax² + bx + c = 0, where a, b, and c are real numbers.

Solving Methods

• Square Root Method: x² - 4 = 0 → x² = 4 → x = ±2. Solution Set (SS) = {2, -2}.
• Completing the Square: For x² + 4x + 10 = 0, add (b/2)² to both sides.
• Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a

Radical Equations

Key Idea: Isolate the radical and square both sides. Always check for extraneous solutions by plugging answers back into the original equation, especially with even-indexed radicals. Odd-indexed radicals do not produce extraneous solutions.