Math Problem Statement

Multiply and simplify the following complex numbers: (2 + 5i) • (5 - 3i)

Solution

The problem shown is:

(2+5i)(53i)(2 + 5i) \cdot (5 - 3i)

To multiply these complex numbers, we'll use the distributive property (FOIL method).

Step-by-step solution:

  1. Multiply the real and imaginary parts: (2)(5)+(2)(3i)+(5i)(5)+(5i)(3i)(2)(5) + (2)(-3i) + (5i)(5) + (5i)(-3i)

  2. Perform each multiplication:

    • 25=102 \cdot 5 = 10
    • 2(3i)=6i2 \cdot (-3i) = -6i
    • 5i5=25i5i \cdot 5 = 25i
    • 5i(3i)=15i25i \cdot (-3i) = -15i^2
  3. Simplify using i2=1i^2 = -1:

    • 15i2=15(1)=15-15i^2 = -15(-1) = 15
  4. Combine all the terms: 10+(6i)+25i+1510 + (-6i) + 25i + 15 Simplifies to: (10+15)+(6i+25i)=25+19i(10 + 15) + (-6i + 25i) = 25 + 19i

Final Answer:

25+19i25 + 19i

Would you like further clarification or details?

Related questions:

  1. What is the general form of a complex number?
  2. How do you divide complex numbers?
  3. How can you represent complex numbers geometrically?
  4. What is the conjugate of a complex number and how is it used?
  5. How do you find the magnitude of a complex number?

Tip:

When multiplying complex numbers, remember to simplify i2i^2 as 1-1 to get a real number.

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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

Imaginary Unit Properties: i^2 = -1

Suitable Grade Level

Grades 10-12