What is (x-yi)/(x+yi+1) ?

The solution to (x-yi)/(x+yi+1) is (x(x+1)-y^2)/((x+1)^2+y^2)+(-2xy-y)/((x+1)^2+y^2)i

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simplify (x-yi)/(x+yi+1)

simplify

\frac{x - y i}{x + y i + 1}

\frac{x ( x + 1 ) - y ^{2}}{( x + 1 ) ^{2} + y ^{2}} + \frac{- 2 x y - y}{( x + 1 ) ^{2} + y ^{2}} i

Solution steps

\frac{x - y i}{x + y i + 1}

Rewrite

x + y i + 1

in standard complex form:

(x + 1) + y i

= \frac{x - y i}{( x + 1 ) + y i}

Multiply by the conjugate

\frac{( x + 1 ) - y i}{( x + 1 ) - y i}

= \frac{( x - y i ) ( ( x + 1 ) - y i )}{( ( x + 1 ) + y i ) ( ( x + 1 ) - y i )}

Simplify

\frac{( x - y i ) ( ( x + 1 ) - y i )}{( ( x + 1 ) + y i ) ( ( x + 1 ) - y i )} : \frac{x ( x + 1 ) - y ^{2} + ( - 2 x y - y ) i}{( x + 1 ) ^{2} + y ^{2}}

= \frac{x ( x + 1 ) - y ^{2} + ( - 2 x y - y ) i}{( x + 1 ) ^{2} + y ^{2}}

Rewrite in standard complex form:

\frac{x ( x + 1 ) - y ^{2}}{( x + 1 ) ^{2} + y ^{2}} + \frac{- 2 x y - y}{( x + 1 ) ^{2} + y ^{2}} i

= \frac{x ( x + 1 ) - y ^{2}}{( x + 1 ) ^{2} + y ^{2}} + \frac{- 2 x y - y}{( x + 1 ) ^{2} + y ^{2}} i

\frac{2}{7} \times \frac{3}{4}

x^{2} - 12 x - 45

5 x^{2} + 2 x + 5 = 0

6^{2} + b^{2} = 7^{2}

8^{- 2} = \frac{1}{64}

What is (x-yi)/(x+yi+1) ?
The solution to (x-yi)/(x+yi+1) is (x(x+1)-y^2)/((x+1)^2+y^2)+(-2xy-y)/((x+1)^2+y^2)i