sqr(120-22i). Express in form a+bi

Answer 1

√120-22i=a+bi

square both sides

120-22i=a^2-b^2+2abi

the real parts are equal and the imaginary parts are equal

120=a^2-b^2 and -22=2ab

-22=2ab

b=-22/2a=-11/a

a^2-b^2=120

a^2-(-11/a)^2=120

a^2-(121/a^2)-120=0

multiply both sides by a^2

a^4-120a^2-121=0

factor

(a^2-121)(a^2+1)=0

a^2-121=0

(a+11)(a-11)=0

a=11 and a=-11

from above b=-11/a

a=11, b=-1

a=-11 and b=1

there are two answers: 11-i and -11+i

now you also have a^2+1=0

a^2=-1

a=i and a=-i

if you follow the same procedure as above you will get the same two answers as we already have; try it if you like

again the answers are 11-i and -11+i

Answer 2

sqr(120-22i). Express in form a+bi

Set X=120-22i

|X|=[(120)²+(-22)²]^1/2

|X|=122

Θ=tan^-1(-22/120)=-0.1813 rad

so X=122e^-0.1813i

Finally, √X=(122e^-0.1813i)^1/2=√122e^-0.0906i=√122[cos(-0.0906)+isin(-0.0906)]

=√122[0.9958-0.0915i]=10.998-1.011i

Since the input data was limited to two significant figures,

√(120-22i)=11-1.0i