Question

EXAMPLE 1 Evaluate V-5 + 12i​

Answer 1

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Answer 2

$\textbf{\underline{\underline{According\:to\:the\:Question}}}$

Assumption

√-5 + 12i = (x + iy) ..... (1)

$\textbf{\underline{Squarring\;both\;sides :-}}$

(-5 + 12i) = (x + iy)²

(-5 + 12i) = (x² - y²) + i(2xy) .... (2)

Here

$\Large{\boxed{\sf\:{Real\;and\;imaginary\;part\;of\;both\;sides\;of\;(2)}}}$

(x² - y²) = -5

2xy = 12

(x + y)² = √{(x² - y²)² + 4x²y²}

= √(25 + 144)

= √169

= 13

Hence,

(x² - y²) = -5 .... (3)

Also,

(x² + y²) = 13 ..... (4)

$\Large{\boxed{\sf\:{Solving\;(3)\;and\;(4)}}}$

x² = 4

x = √4

x = 2

Also,

y² = 9

y = √9

y = 3

Here

xy > 0

Therefore,

(x = 2 and y = 3) or (x = -2 and y = -3)

Hence,

√-5 + 12i = (2 + 3i) or (-2 - 3i)