Find the square roots of 16 - 30i

Let √( 16 - 30i ) = x - iy ---( 1 )

Do the square of equation ( 1 ) , we

get ,

16 - 30i = ( x - iy )²

= x² + ( iy )² - 2 × x × iy

= ( x² - y² ) - 2xyi

Compare both sides ,

x² - y² = 16 ---( 2 )

2xy = 30

y = 30/2x

y = 15/x ---( 3 )

Substitute y value in equation ( 2 ),

we get

x² - ( 15/x )² = 16

x² - 225/x² = 16

x⁴ - 225 = 16x²

x⁴ - 16x² - 225 = 0

x⁴ - 25x² + 9x² - 225 = 0

x² ( x² - 25 ) + 9 ( x² - 25 ) = 0

( x² - 25 ) ( x² + 9 ) = 0

x² - 25 = 0 or x² + 9 = 0

x² = 25

x = ± 5

Substitute x = ± in equation ( 3 ) ,

We get ,

y = 15/( ± 5 )

y = ± 3

Therefore ,

x = ± 5 , y = ± 3

√( 16 - 30 i ) = x - iy

= ± ( 5 - 3i )

f(x) has value 150 at x= 0

and 0 at x= 3

so d(f(x))/dt = slope = -150/3 = -50

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