Question

If the square roots of 3 + 4i is $\pm (a+ib)$. Find the value of a + b.

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Answer 1

Let z = 3 + 4i
Square root of 3 + 4i = a + ib
$\sqrt{3+4i}=a+ib\dots \dots$ (1)
Square on both sides
3 + 4i = $(a+ib{)}^2=a^2-b^2+2abi$
Compare real part and imaginary part on both sides
$a^2-b^2=3\dots \dots$ (2)
2abi = 4
$ab=2\dots \dots$ (3)

Solving equation 2 and equation 4
$a^2-b^2$ = 3
$a^2+b^2$ = 5
2$a^2$ = 8
a = $\pm$ 2
Substituting value of a in equation 4
$b^2$ = 1
b = $\pm$ 1
Square root of 3 + 4i = $\pm$ (2 + i) = $\pm$(2 + i)
Since imaginary part of 3 + 4i is positive, imaginary part of square root {$\pm$(2 + i)} should also be positive.
a + b = 2 + 1 = 3