Find-square-root-of-7-30-2-i-

Question Number 27681 by ajfour last updated on 12/Jan/18

F i n d s q u a r e r o o t o f 7 - 30 \sqrt{2} i .

Commented by Rasheed.Sindhi last updated on 13/Jan/18

Squareroot of 7 - 30 \sqrt{2} i ?

Let \pm \sqrt{7 - 30 \sqrt{2} i} = p + q \sqrt{2} i

where p, q \in Q

{(\pm \sqrt{7 - 30 \sqrt{2} i})}^{2} = {(p + q \sqrt{2} i)}^{2}

7 - 30 \sqrt{2} i = p^{2} - {2 q}^{2} + 2 pq \sqrt{2} i

p^{2} - {2 q}^{2} = 7 \land 2 pq = - 30

q = - \frac{15}{p}

p^{2} - 2 {(- \frac{15}{p})}^{2} = 7

p^{2} - \frac{450}{p^{2}} = 7

p^{4} - {7 p}^{2} - 450 = 0

p^{2} = \frac{7 \pm \sqrt{49 + 1800}}{2}

= \frac{7 \pm 43}{2} = 25, - 18

p = \pm 5 or p = \pm 3 \sqrt{2} i \notin Q (discardable)

q = - \frac{15}{\pm 5} = \mp 3

p + q \sqrt{2} i = \pm 5 \mp 3 \sqrt{2} i

= 5 - 3 \sqrt{2} i or - 5 + 3 \sqrt{2} i

Squareroot of 7 - 30 \sqrt{2} i are

5 - 3 \sqrt{2} i or - 5 + 3 \sqrt{2} i

Commented by Tinkutara last updated on 13/Jan/18

Thank you very much Sir! I got the answer.