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Question

Mathematics Question on Complex Numbers and Quadratic Equations

Express 5+2i1−2i\frac{5+\sqrt{2}i}{1-\sqrt{2}i} in the form of a+iba + ib .

A

1−22i1-2 \sqrt{2}i

B

1−2i1-\sqrt{2}i

C

1+22i1+2\sqrt{2}i

D

1+2i1+\sqrt{2}i

Answer

1+22i1+2\sqrt{2}i

Explanation

Solution

We have, 5+2i1−2i\frac{5+\sqrt{2}i}{1-\sqrt{2}i} =5+2i1−2i×1+2i1+2i=\frac{5+\sqrt{2}i}{1-\sqrt{2}i}\times\frac{1+\sqrt{2}i}{1+\sqrt{2}i} =5+52i+2i−21−(2i)2=\frac{5+5\sqrt{2}i+\sqrt{2}i-2}{1-\left(\sqrt{2}i\right)^{2}} =3(1+22i)3=\frac{3\left(1+2 \sqrt{2}i\right)}{3} =1+22i=1+2\sqrt{2}i