(1+3i)z^2+(5+6i)z+7-6i=0

解

(1 + 3 i) z^{2} + (5 + 6 i) z + 7 - 6 i = 0

z = \frac{- 5.46069 \dots + 1.2 E - 14}{2} - 0.07678 \dots i, z = \frac{- 5.46069 \dots - 1.2 E - 14}{2} - 0.07678 \dots i, z = \frac{0.86069 \dots + 1.11 E - 14}{2} + 0.97678 \dots i, z = \frac{0.86069 \dots - 1.11 E - 14}{2} + 0.97678 \dots i

解答ステップ

(1 + 3 i) z^{2} + (5 + 6 i) z + 7 - 6 i = 0

代用

z = x + y i

(1 + 3 i) (x + y i)^{2} + (5 + 6 i) (x + y i) + 7 - 6 i = 0

拡張

(1 + 3 i) (x + y i)^{2} + (5 + 6 i) (x + y i) + 7 - 6 i : (x^{2} - y^{2} - 6 x y + 5 x - 6 y + 7) + i (- 6 + 3 x^{2} - 3 y^{2} + 5 y + 6 x + 2 x y)

(x^{2} - y^{2} - 6 x y + 5 x - 6 y + 7) + i (- 6 + 3 x^{2} - 3 y^{2} + 5 y + 6 x + 2 x y) = 0

標準的な複素数形式で

0

を書き換える:

0 + 0 i

(x^{2} - y^{2} - 6 x y + 5 x - 6 y + 7) + i (- 6 + 3 x^{2} - 3 y^{2} + 5 y + 6 x + 2 x y) = 0 + 0 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} - 6 x y + 5 x - 6 y + 7 = 0 - 6 + 3 x^{2} - 3 y^{2} + 5 y + 6 x + 2 x y = 0]

[x^{2} - y^{2} - 6 x y + 5 x - 6 y + 7 = 0 - 6 + 3 x^{2} - 3 y^{2} + 5 y + 6 x + 2 x y = 0] : x = \frac{- 5.46069 \dots + 1.2 E - 14}{2}, x = \frac{- 5.46069 \dots - 1.2 E - 14}{2}, x = \frac{0.86069 \dots + 1.11 E - 14}{2}, x = \frac{0.86069 \dots - 1.11 E - 14}{2}, x = \frac{- 5.46069 \dots + 1.2 E - 14}{2}, x = \frac{- 5.46069 \dots - 1.2 E - 14}{2}, x = \frac{0.86069 \dots + 1.11 E - 14}{2}, x = \frac{0.86069 \dots - 1.11 E - 14}{2}, y = - 0.07678 \dots y = - 0.07678 \dots y = 0.97678 \dots y = 0.97678 \dots y = - 0.07678 \dots y = - 0.07678 \dots y = 0.97678 \dots y = 0.97678 \dots

代用を戻す

z = x + y i

z = \frac{- 5.46069 \dots + 1.2 E - 14}{2} - 0.07678 \dots i, z = \frac{- 5.46069 \dots - 1.2 E - 14}{2} - 0.07678 \dots i, z = \frac{0.86069 \dots + 1.11 E - 14}{2} + 0.97678 \dots i, z = \frac{0.86069 \dots - 1.11 E - 14}{2} + 0.97678 \dots i