Question

Calculate an estimate of a square root of 119 + 120i

Answer 1

Explanation:

Given

z=119+120 i

Let
`√(119+120 i)=p+iq`

Squaring both sides

`119+120 i=p^2-q^2+2ipq`

Comparing real and imaginary part

Re(LHS)=Re(RHS)

`119=p^2-q^2`-----------1

comparing Im(LHS)=Im(RHS)

120=2pq

`q=(60)/(p)`

Substitute q in 1

`119=p^2-((60)/(p))^2`

`p^4-119p^2-(68)^2=0`

Let
`x=p^2`

`x^2-119x-4624=0`

`x=frac{119\pm √(119^2+4* 4624)}{2}`

`x=(119\pm 180.71)/(2)`

we take only Positive value because
`p^2=x`

x=149.85

`p^2=149.85`

thus
`p=\pm 12.24`

`q=\mp 4.90`

thus
`√(119+120 i)=\pm (12.24+i 4.90)`