simplify (x-1+iy)^2

Solution

simplify

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

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

Solution steps

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

Apply exponent rule:

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

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

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

Distribute parentheses

= xx + x (- 1) + x i y - 1 \cdot x - 1 \cdot (- 1) - 1 i y + i y x + i y (- 1) + i y i y

Simplify

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

= x^{2} - 2 x - 2 y i + 2 x y i - 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{- 2 x ^{2} - 22 x - 2}{3 ( x - 2 ) ( x + 1 ) ( x - 5 )}

(x + \frac{7}{2})^{2}

e^{y + C} ∣ y ∣^{3} (4 - 4 x + y)

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

(x + \frac{5}{3}) (x - \frac{1}{3})