SQUARE ROOT OF A COMPLEX NUMBER

Example 1 :

Find the square root of (-7 + 24i).

Solution :

Square both sides.

-7 + 24i = x² + 2xyi + (yi)²

-7 + 24i = x² + 2x(yi) + y²i²

-7 + 24i = x² + 2xyi + y²(-1)

-7 + 24i = x² + 2xyi - y²

-7 + 24i = (x² - y²) + 2xyi

Equate the real and imaginary parts.

Substitute y = ¹²⁄ₓ into (1) and solve for x.

x⁴ - 144 = -7x²

x⁴ + 7x² - 144 = 0

(x²)² + 7x² - 144 = 0

Let y = x².

y² + 7y - 144 = 0

Factor and solve for y.

y² - 9y + 16y - 144 = 0

y(y - 9) + 16(y - 9) = 0

(y - 9)(y + 16) = 0

y - 9 = 0  or  y + 16 = 0

y = 9  or y = -16

Replace y by x².

So, x = -3 or x = 3.

Substiute x = -3 and x = 3 into (2).

Since xy = 12, both the values of x and y must have the same sign.

(x, y) = (-3, -4)  or  (3, 4)

Therefore,

Example 2 :

Find the square root of (-8 - 6i).

Solution :

Square both sides.

-8 - 6i = x² + 2xyi + (yi)²

-8 - 6i = x² + 2x(yi) + y²i²

-8 - 6i = x² + 2xyi + y²(-1)

-8 - 6i = x² + 2xyi - y²

-8 - 6i = (x² - y²) + 2xyi

Equate the real and imaginary parts.

Substitute y = ⁻³⁄ₓ into (1) and solve for x.

x⁴ - 9 = -8x²

x⁴ + 8x² - 9 = 0

(x²)² + 8x² - 9 = 0

Let y = x².

y² + 8y - 9 = 0

Factor and solve for y.

y² - y + 9y - 9 = 0

y(y - 1) + 9(y - 1) = 0

(y - 1)(y + 9) = 0

y - 1 = 0  or  y + 9 = 0

y = 1  or y = -9

Replace y by x².

So, x = -1 or x = 1.

Substiute x = -1 and x = 1 into (2).

Since xy = -3, the values of x and y must have different signs.

(x, y) = (-1, 3)  or  (1, -3)

Therefore,

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