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Multiply and simplify the following complex numbers: (1+i)*(3-5i) +x

Multiply and simplify the following complex numbers:\newline(1+i)⋅(3−5i) (1+i) \cdot(3-5 i)

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Full solution

Q. Multiply and simplify the following complex numbers:\newline(1+i)⋅(3−5i) (1+i) \cdot(3-5 i)
  1. Distribute terms: Distribute each term in the first complex number by each term in the second complex number.\newline(1+i)⋅(3−5i)=1⋅(3−5i)+i⋅(3−5i)(1+i)\cdot(3-5i) = 1\cdot(3-5i) + i\cdot(3-5i)
  2. Multiply real and imaginary parts: Multiply the real parts and the imaginary parts separately.\newline1⋅(3−5i)=3−5i1\cdot(3-5i) = 3 - 5i\newlinei⋅(3−5i)=3i−5i2i\cdot(3-5i) = 3i - 5i^2
  3. Simplify using i2i^2: Remember that i2i^2 is equal to −1-1.\newline3i−5i2=3i−5(−1)3i - 5i^2 = 3i - 5(-1)
  4. Combine like terms: Simplify the expression by substituting i2i^2 with −1-1 and combining like terms.\newline3i−5(−1)=3i+53i - 5(-1) = 3i + 5
  5. Add results: Add the results from Step 11 and Step 44 to get the final product.\newline(3−5i)+(3i+5)=3+5+3i−5i(3 - 5i) + (3i + 5) = 3 + 5 + 3i - 5i
  6. Combine real and imaginary parts: Combine the real parts and the imaginary parts.\newline3+5+3i−5i=8+(3i−5i)3 + 5 + 3i - 5i = 8 + (3i - 5i)
  7. Simplify imaginary parts: Simplify the imaginary parts. 8+(3i−5i)=8−2i8 + (3i - 5i) = 8 - 2i

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