Math Problem Statement

Diketahui (x + 3i) × (2 - 2i) = 4 + 8i. Nilai 3x adalah ...

Solution

To solve the equation:

(x+3i)(22i)=4+8i(x + 3i) \cdot (2 - 2i) = 4 + 8i

Step 1: Expand the left-hand side

Using the distributive property: (x+3i)(22i)=2x2xi+6i6i2(x + 3i)(2 - 2i) = 2x - 2xi + 6i - 6i^2

Recall that i2=1i^2 = -1, so: 6i2=6-6i^2 = 6

This simplifies to: 2x2xi+6i+62x - 2xi + 6i + 6

Group real and imaginary terms: (2x+6)+(2x+6)i(2x + 6) + (-2x + 6)i

Step 2: Set the real and imaginary parts equal

The equation becomes: (2x+6)+(2x+6)i=4+8i(2x + 6) + (-2x + 6)i = 4 + 8i

From the real part: 2x+6=42x + 6 = 4

From the imaginary part: 2x+6=8-2x + 6 = 8

Step 3: Solve for xx

From the real part: 2x=462x = 4 - 6 2x=2    x=12x = -2 \implies x = -1

Check the imaginary part: 2(1)+6=8-2(-1) + 6 = 8 2+6=8(True!)2 + 6 = 8 \quad \text{(True!)}

Thus, x=1x = -1.

Step 4: Calculate 3x3x

3x=3(1)=33x = 3(-1) = -3

Final Answer:

3\boxed{-3}

Let me know if you'd like more details or have further questions! Here are some related questions to explore:

  1. How is i2=1i^2 = -1 derived and used in complex number calculations?
  2. What is the geometric interpretation of multiplying complex numbers?
  3. How do you simplify complex number expressions?
  4. What are some applications of complex numbers in engineering or physics?
  5. How can you solve similar equations with unknowns in both parts of a complex number?

Tip: Always separate real and imaginary parts carefully to avoid errors when solving complex equations.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Algebra
Distributive Property

Formulas

(a + bi)(c + di) = (ac - bd) + (ad + bc)i

Theorems

Properties of Complex Numbers

Suitable Grade Level

Grades 10-12