simplify (x+iy+1)^2

Solution

simplify

(x + i y + 1)^{2}

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

Solution steps

(x + i y + 1)^{2}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

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

= (x + i y + 1) (x + i y + 1)

Distribute parentheses

= xx + x i y + x \cdot 1 + i y x + i y i y + i y \cdot 1 + 1 \cdot x + 1 i y + 1 \cdot 1

Simplify

xx + x i y + x \cdot 1 + i y x + i y i y + i y \cdot 1 + 1 \cdot x + 1 i y + 1 \cdot 1 : x^{2} + 2 y i + 2 x y i + 2 x - y^{2} + 1

= x^{2} + 2 y i + 2 x y i + 2 x - y^{2} + 1

Rewrite in standard complex form:

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

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

\frac{6 x ^{2}}{( x ^{2} + 3 ) ^{2}}

\frac{- 3 x - 7}{( x + 1 ) ^{3}}

(1 + cot (θ)) (1 - tan (θ))

(4 x - \frac{2}{3}) (4 x + \frac{2}{3})

\frac{l}{s ( s + 2 )}