Complex Numbers and Quadratic Equations

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Step 1: Rearrange to standard form → x² = -1 Step 2: Take square root of both sides → x = ±√(-1) Step 3: Since √(-1) = i → x = ±i Answer: x = i or x = -i Note: This shows why we need complex numbers -…

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Step 1: Group real parts and imaginary parts separately Real parts: 3 + 2 = 5 Imaginary parts: 4i + (-7i) = -3i Step 2: Combine results Answer: 5 - 3i Rule: (a + bi) + (c + di) = (a + c) + (b + d)i…

Answer

Step 1: Use FOIL method = 2(4) + 2(-5i) + 3i(4) + 3i(-5i) = 8 - 10i + 12i - 15i² Step 2: Remember i² = -1 = 8 - 10i + 12i - 15(-1) = 8 - 10i + 12i + 15 Step 3: Combine like terms = (8 + 15) + (-10 + 1…

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Conjugate z̄: Step 1: Change sign of imaginary part z̄ = 5 - (-12i) = 5 + 12i Modulus |z|: Step 2: Use formula |z| = √(a² + b²) |z| = √(5² + (-12)²) = √(25 + 144) = √169 = 13 Answers: z̄ = 5 + 12i, …

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Step 1: Use the pattern of powers of i i¹ = i, i² = -1, i³ = -i, i⁴ = 1 Pattern repeats every 4 powers Step 2: Divide exponent by 4 47 ÷ 4 = 11 remainder 3 So i⁴⁷ = i³ Step 3: From the pattern i³ = …

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Step 1: Multiply numerator and denominator by conjugate of denominator (6 + 8i)/(3 - 4i) × (3 + 4i)/(3 + 4i) Step 2: Multiply numerator (6 + 8i)(3 + 4i) = 18 + 24i + 24i + 32i² = 18 + 48i - 32 = -14 …

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Step 1: Factor out the negative sign √(-16) = √(-1 × 16) = √(-1) × √(16) Step 2: Use definition √(-1) = i = i × √(16) = i × 4 Step 3: Simplify = 4i Answer: √(-16) = 4i General rule: √(-a) = i√(a) …

Answer

Step 1: Identify coefficients a = 1, b = -4, c = 5 Step 2: Calculate discriminant D = b² - 4ac = (-4)² - 4(1)(5) = 16 - 20 = -4 Step 3: Since D < 0, solutions are complex x = [-b ± √D]/2a = [4 ± √(-…