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Multiply and simplify the following complex numbers: (1+2i)*(1-4i) -x +1

Multiply and simplify the following complex numbers:\newline(1+2i)⋅(1−4i) (1+2 i) \cdot(1-4 i)

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Multiply numbers written in scientific notationright chevron icon

Full solution

Q. Multiply and simplify the following complex numbers:\newline(1+2i)⋅(1−4i) (1+2 i) \cdot(1-4 i)
  1. Apply distributive property: Apply the distributive property (also known as the FOIL method for binomials) to multiply the complex numbers.\newline(1+2i)⋅(1−4i)=1⋅(1)+1⋅(−4i)+2i⋅(1)+2i⋅(−4i)(1+2i)\cdot(1-4i) = 1\cdot(1) + 1\cdot(-4i) + 2i\cdot(1) + 2i\cdot(-4i)
  2. Perform multiplication for each term: Perform the multiplication for each term.\newline1×(1)=11\times(1) = 1\newline1×(−4i)=−4i1\times(-4i) = -4i\newline2i×(1)=2i2i\times(1) = 2i\newline2i×(−4i)=−8i22i\times(-4i) = -8i^2 (Remember that i2=−1i^2 = -1)
  3. Substitute i2i^2 and simplify: Substitute i2i^2 with −1-1 and simplify the expression.1+(−4i)+2i+(−8×−1)=1−4i+2i+81 + (-4i) + 2i + (-8 \times -1) = 1 - 4i + 2i + 8
  4. Combine like terms: Combine like terms.\newline(1+8)+(−4i+2i)=9−2i(1 + 8) + (-4i + 2i) = 9 - 2i

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