(−2+4i)⋅(5+i)

Jasonbradley

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Jun 17, 2024

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Multiplying Complex Numbers: (−2+4i)⋅(5+i)

This article will guide you through the process of multiplying two complex numbers: (-2 + 4i) and (5 + i).

Understanding Complex Numbers

Complex numbers are numbers that can be expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as the square root of -1 (i² = -1).

Multiplication Process

To multiply complex numbers, we follow the same distributive property used for multiplying binomials:

1. FOIL Method: This stands for "First, Outer, Inner, Last". We multiply each term in the first complex number by each term in the second complex number:

   • First: (-2) * 5 = -10
   • Outer: (-2) * i = -2i
   • Inner: 4i * 5 = 20i
   • Last: 4i * i = 4i²

2. Simplify: Remember that i² = -1. Substitute this value and combine the real and imaginary terms:

   • -10 - 2i + 20i + 4(-1)
   • -10 - 4 + 18i

3. Final Result: Combine the real terms and the imaginary terms:

   • (-2+4i)⋅(5+i) = -14 + 18i

Conclusion

Therefore, the product of (-2 + 4i) and (5 + i) is -14 + 18i. By understanding the properties of complex numbers and applying the distributive property, we can effectively multiply complex numbers.