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 \cdot (- 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 \cdot (- 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

- (x - 1) (x - 1 + 3) (x - 1 - 3)

\frac{2 ( π - 2 2 )}{3 π - 8}

\frac{5 ( - x ^{2} + 36 )}{( x ^{2} + 36 ) ^{2}}

6 e^{(- 2 x^{2} + 5 x)} \cdot (- 4 x + 5)

(4 x y - y^{2}) d x + (2 x^{2} - 2 x y)