Question

$\frac{3 + 4i}{7 + 2i} =$
Simplify. 'i' is an imaginary number.
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Answer 1

Answer:

$\large\boxed{\dfrac{29+22i}{53}=\dfrac{29}{53}+\dfrac{22}{53}i}$

Step-by-step explanation:

$\dfrac{3+4i}{7+2i}\cdot\dfrac{7-2i}{7-2i}\qquad\text{use}\ (a+b)(a-b)=a^2-b^2\\\\=\dfrac{(3+4i)(7-2i)}{(7+2i)(7-2i)}\qquad\text{use FOIL}\ (a+b)(c+d)=ac+ad+bc+bd\\\\=\dfrac{(3)(7)+(3)(-2i)+(4i)(7)+(4i)(-2i)}{7^2-(2i)^2}\\\\=\dfrac{21-6i+28i-8i^2}{49-2^2i^2}\qquad\text{use}\ \sqrt{-1}=i\to i^2=-1\\\\=\dfrac{21-6i+28i-8(-1)}{49-(4)(-1)}$

$=\dfrac{21-6i+28i+8}{49+4}\qquad\text{combine like terms}\\\\=\dfrac{(21+8)+(-6i+28i)}{53}\\\\=\dfrac{29+22i}{53}\\\\=\dfrac{29}{53}+\dfrac{22}{53}i$