z^2=49+i

解

z^{2} = 49 + i

z = \frac{49 + 2402}{2} + \frac{1}{2 49 + 2402} i, z = - \frac{49 + 2402}{2} - \frac{1}{2 49 + 2402} i

解答ステップ

z^{2} = 49 + i

代用

z = x + y i

(x + y i)^{2} = 49 + i

拡張

(x + y i)^{2} : (x^{2} - y^{2}) + 2 i x y

(x^{2} - y^{2}) + 2 i x y = 49 + i

複素数は, その実数部と虚数部が等しい場合にのみ等しくなるequation系として書き換える:

[x^{2} - y^{2} = 49 2 x y = 1]

[x^{2} - y^{2} = 49 2 x y = 1] : x = \frac{49 + 2402}{2}, x = - \frac{49 + 2402}{2}, y = \frac{1}{2 49 + 2402} y = - \frac{1}{2 49 + 2402}

代用を戻す

z = x + y i

z = \frac{49 + 2402}{2} + \frac{1}{2 49 + 2402} i, z = - \frac{49 + 2402}{2} - \frac{1}{2 49 + 2402} i