Question

find the square root of 4+6√-5 please solve completely​

Answer 1

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suppose the root of 4+6√-5 is A+Bi

√(4+6√5i) = A+Bi

4+6√5 = A^2 - B^2 + 2ABi

therefore

A^2 - B^2 = 4. | 6√5i = 2ABi

A^2 - (3√5/A)^2 = 4 | 3√5 = AB

A^2 -(9×5/A^2) = 4. | 3√5/A = B

A^4 - 45 = 4A^2.

A^4 - 4A^2 - 45 = 0

(A^2)^2 - 2×2×A^2 - 4 + 4 - 45 =0

(A^2 - 2)^2 - 49 = 0. | If A = 3 then B = √5

(A^2 - 2)^2 = 49. | If A = -3 then B = -√5

A^2 - 2 = 7.

A^2 = 9

A = +- 3

therefore roots of √(4+i6√5) are 3+i√5 ,-3-i√5

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