Math Problem Statement

(-7 + 3i)(-2 + 6i)

Solution

The problem shown in the image is:

(7+3i)(2+6i)(-7 + 3i)(-2 + 6i)

To simplify this, we need to apply the distributive property (also known as the FOIL method for binomials). Here’s the process step by step:

Step 1: Distribute the terms

(7+3i)(2+6i)=(7)(2)+(7)(6i)+(3i)(2)+(3i)(6i)(-7 + 3i)(-2 + 6i) = (-7)(-2) + (-7)(6i) + (3i)(-2) + (3i)(6i)

Step 2: Multiply the terms

  • (7)(2)=14(-7)(-2) = 14
  • (7)(6i)=42i(-7)(6i) = -42i
  • (3i)(2)=6i(3i)(-2) = -6i
  • (3i)(6i)=18i2(3i)(6i) = 18i^2

Step 3: Simplify

Remember that i2=1i^2 = -1. So: 18i2=18(1)=1818i^2 = 18(-1) = -18

Now, add all the terms: 1442i6i1814 - 42i - 6i - 18

Step 4: Combine like terms

(1418)+(42i6i)=448i(14 - 18) + (-42i - 6i) = -4 - 48i

Thus, the simplified answer is: 448i-4 - 48i

Would you like further details or explanations?

Here are 5 related questions you might consider:

  1. How do you multiply two complex numbers in general form?
  2. What is the significance of i2=1i^2 = -1 in complex number operations?
  3. How would the process differ if there were more than two complex numbers to multiply?
  4. What are some applications of complex numbers in real-life problems?
  5. How can you represent the result of this multiplication graphically on the complex plane?

Tip: Always remember that i2=1i^2 = -1, and it plays a critical role when simplifying products involving complex numbers.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Algebraic Multiplication
Imaginary Numbers

Formulas

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

Theorems

Definition of Complex Multiplication
i^2 = -1

Suitable Grade Level

Grade 10-12 (High School)