SOLUTION: What is the solution to {{{(2+5i)/(5+2i)}}}?

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Question 122353: What is the solution to %282%2B5i%29%2F%285%2B2i%29?
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!

What is the solution to %282%2B5i%29%2F%285%2B2i%29?

%282%2B5i%29%2F%285%2B2i%29

Put parentheses around numerator and
denominator:

%28%282%2B5i%29%29%2F%28%285%2B2i%29%29

Form the conjugate of the denominator by
changing the sign of its term with "i".
The conjugate of denominator %285%2B2i%29
is %285-2i%29.
So we multiply by %28%285-2i%29%29%2F%28%285-2i%29%29

%28%282%2B5i%29%29%2F%28%285%2B2i%29%29·%28%285-2i%29%29%2F%28%285-2i%29%29

%28%282%2B5i%29%285-2i%29%29%2F%28%285%2B2i%29%285-2i%29%29

Us FOIL on top and bottom:

%2810-4i%2B25i-10i%5E2%29%2F%2825-10i%2B10i-4i%5E2%29

Combine terms:

%2810%2B21i-10i%5E2%29%2F%2825-4i%5E2%29

replace i%5E2 by -1

%2810%2B21i-10%28-1%29%29%2F%2825-4%28-1%29%29

Simplify:

%2810%2B21i%2B10%29%2F%2825%2B4%29

Combine terms

%2820%2B21i%29%2F%2829%29

Write as two fractions:

20%2F29%2B21%2F29i

Edwin