SOLUTION: find the complex conjugate of (3+5i)/(1+2i)

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Question 423785: find the complex conjugate of (3+5i)/(1+2i)
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!

The complex conjugate of the complex number a+bi is a-bi (or a + (-bi)). To find the complex conjugate of your expression we will start by writing it in a+bi form.

To write your expression in a+bi form, we will... rationalize its denominator!? Remember that i = so your denominator, with its 2i, has a square root! So it needs to be rationalized just like denominators with "regular" square roots like the one in the expression:

To rationalize two-term denominators we take advantage of the pattern. The pattern shows us how to take a two-term expression, a+b or a-b, multiply it by its conjugate and get an expression of perfect squares.

So to rationalize your denominator we multiply the numerator and denominator by the conjugate of its denominator:

We use FOIL to multiply the numerators. Although FOIL can also be used to multiply the denominators, it is faster to use the pattern:

which simplifies as follows:


Since this becomes:

which simplifies further:


Writing this in standard (a+bi) form:

or

This is your original expression in standard form.

Its conjugate would be:

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