Question

find the square root -4-3i

Answer 1

Given:

-4-3i

To find:

Find the square root -4-3i

Solution:

From given, we have,

-4 - 3i

Let the square root of a + ib be p + iq

where, a = -4 and b = -3

Then, we have,

(p + iq)² = a + ib

equating the real and imaginary parts, we have,

p² - q² = a = -4 ....(1)

2pq = b = -3 ......(2)

using (2), we have,

q = -3/2p ....(3)

we use the formula,

p = 1/√2 ×√[a+√(a²+b²)]

p = 1/√2 ×√[(-4)+√((-4)²+(-3)²)]

p = 1/√2 ×√[-4+√(16+9)]

p = 1/√2 ×√[-4+√(25)]

p = 1/√2 ×√[-4+5]

p = 1/√2 ×√[1]

p = 1/√2

using the value of "p" in equation (3), we get,

q = -3/[2 (1/√2)]

q = -3/√2

Therefore, the square root of  -4-3i is 1/√2 - i3/√2 = (1-i3)/√2.

Answer 2

Given  (Z)= -4-3i

The square root of the equation is given by the expression

X = √1/2[lzI+a]

Y=  √1/2 [lzI-a]

|z|= √(16+9) = √25 = 5

Substituting values,

X=  ±√1/2(5-4) ⇒ ±√[1/2(1)]

or, x =± 1/√2 ⇒ ± 1/√2

Y =∓ √{1/2(5+4)} = √{1/2 (9)}

or, Y=∓3 /√2.

∴ square root = ± 1/√2 ∓  3/√2i.