Question

Find the complex number whose square is the complex number
$a + 2 + i \sqrt{3 {a}^{2} - 8a - 3 }$
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Answer 1

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Step-by-step explanation:

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Answer 2

Answer:

Square root of given complex number is +-(√( (3a + 1)/2) + i√( (a - 3)/2) ).

Step-by-step explanation:

• A number of the form z = x + iy, where x, y ∈ R, is called a complex number.
• x is real part of complex number while y is imaginary part of complex number.
• x = Re(z) and y = Im(z).

Square Root of Complex Number :

• If z = x + iy, then

$\sqrt{z} = + - ( \frac{ \sqrt{ |z| } + x }{2} + i \frac{ \sqrt{ |z| - x } }{2} )$

Given that :

$z =( a + 2 )+ i \sqrt{(3 {a}^{2} - 8a - 3) }$

Solution :

• Modulus of given complex number, z is

${ |z| }^{2} = {( a+ 2)}^{2} + (3 {a}^{2} - 8a - 3) \\ { |z| }^{2}= {a}^{2} + 4a + 4 + 3 {a}^{2} - 8a - 3 \\ { |z| }^{2} = 4 {a}^{2} - 4a + 1 \\ { |z| }^{2} = {(2a - 1)}^{2} \\ |z| = 2a - 1$

• The square root of given complex number is

$\sqrt{z} =+- ( \sqrt{ \frac{2a - 1 + a + 2}{2} + i \sqrt{ \frac{2a - 1 - a - 2}{2} } } )\\ = +-( \sqrt{ \frac{3a + 1}{2} } + i \sqrt{ \frac{a - 3}{2} })$

• Hence, square root of given complex number is +-(√( (3a + 1)/2) + i√( (a - 3)/2) ) .