Empma u2 ecuaciones cubicas

𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑 = 0 ,

𝑥 =𝑦−

𝑦 3 + 𝑝𝑦 + 𝑞 = 0 , 𝑦

𝑝=𝑐−

𝑏2 , 3

𝑞=𝑑−

𝑝

𝑞

𝑏𝑐 2𝑏 3 + .

3 27

𝑦1 , 𝑦2 , 𝑦3

𝑥1 = 𝑦1 −

𝑏 , 3

𝑥2 = 𝑦2 −

𝑏 , 3

𝑦 3 + 𝑝𝑦 + 𝑞 = 0

𝑦=𝑧−

𝑝 , 3𝑧

𝑏 𝑥3 = 𝑦3 − .

3

𝑏 3

𝑧3 −

𝑝3 27𝑧 3

+𝑞 =0

𝑧3 𝑧 6 + 𝑞𝑧 3 −

𝑝3 = 0.

27 𝑡 = 𝑧3

𝑡 2 + 𝑞𝑡 −

𝑞 𝑡 = 𝑧 3 = − ± √𝑅, 2

𝑝3 =0, 27

𝑞 2 𝑝 3 𝑅 =( ) +( ) .

2 3

1 1 𝜔 = − + 𝑖√3, 2 2

1 1 𝜔2 = − − 𝑖√3 , 2 2

𝑞 3 𝐴 = √− + √𝑅 , 2

𝑞 3 𝐵 = √− − √𝑅 , 2

𝑝

𝐴𝐵 = − 3 𝑝 3

(− 3)

𝑧 𝐴, 𝜔𝐴, 𝜔2 𝐴, 𝐵, 𝜔𝐵, 𝜔2 𝐵. 𝑝

−3

𝑝 𝐴𝐵 = − , 3

𝑝 𝜔𝐴 ∙ 𝜔2 𝐵 = − , 3

𝑝 𝜔2 𝐴 ∙ 𝜔𝐵 = − .

3

𝑧

− 𝑦.

𝑝 3𝑧

𝑦=𝑧−

𝑝 3𝑧

𝑦 𝑦2 = 𝜔𝐴 + 𝜔2 𝐵,

𝑦1 = 𝐴 + 𝐵,

𝑦3 = 𝜔2 𝐴 + 𝜔𝐵 ,

𝑞 2 𝑝3 Δ =√ , √𝑅 = √ + 4 27 108 Δ = 4𝑝3 + 27𝑞2

Δ>0

Δ

√𝑅 𝑦1 𝑦2 , 𝑦3 𝑦1 = 𝐴 + 𝐵 , 𝑦2 = 𝜔𝐴 + 𝜔2 𝐵 , 𝑦2 = 𝜔2 𝐴 + 𝜔𝐵 .

Δ<0

√𝑅

𝑦1 = 2𝑎 , 𝑦2 = −𝑎 − 𝑏√3 , 𝑦2 = −𝑎 + 𝑏√3 . 𝑎

𝑏 3

√𝐴 = 𝑎 + 𝑖𝑏

Δ=0

𝑞 𝐴 = 𝐵 = − ⁄2

3

√𝐵 = 𝑎 − 𝑖𝑏

3 𝑞 𝑦1 = 2√− ⁄2 , 3 𝑞 𝑦2 = 𝑦3 = 2√ ⁄2 .

𝑞=0

𝑥 3 + 6𝑥 2 + 2𝑥 + 12 = 0 . 𝑏 = 6, 𝑐 = 2

𝑑 = 12

𝑏2 62 𝑝=𝑐− =2− = −10, 3 3

𝑝

𝑞

𝑏𝑐 2𝑏 3 6 ∙ 2 2 ∙ 63 𝑞=𝑑− + = 12 − + = 24, 3 27 3 27

𝑦 3 − 10𝑦 + 24 = 0 , Δ Δ = 4𝑝3 + 27𝑞2 = 4 ∙ (−10)3 + 27 ∙ 242 = 11552 , Δ

√𝑅 = √108 𝑦1 𝑦2 , 𝑦3

3 𝑞 24 2888 3 −108 + 38√6 3 𝐴 = √− + √𝑅 = √− + √ =√ , 2 2 27 9

3 𝑞 24 2888 3 −108 − 38√6 𝐵 = √− − √𝑅 = √− − √ =√ .

2 2 27 9 3

3 −108 + 38√6 3 −108 − 38√6 𝑦1 = 𝐴 + 𝐵 = √ +√ = −4 .

9 9

3 3 1 1 −108 + 38√6 1 1 −108 − 38√6 𝑦2 = 𝜔𝐴 + 𝜔2 𝐵 = (− + 𝑖√3) √ + (− − 𝑖√3) √ , 2 2 9 2 2 9 1 1 = (1 − 𝑖√3)(6 − √6) + (1 + 𝑖√3)(6 + √6) , 6 6 = 2 + 𝑖√2 .

3 3 1 1 −108 + 38√6 1 1 −108 − 38√6 𝑦2 = 𝜔2 𝐴 + 𝜔𝐵 = (− − 𝑖√3) √ + (− + 𝑖√3) √ , 2 2 9 2 2 9 1 1 = (1 + 𝑖√3)(6 − √6) + (1 − 𝑖√3)(6 + √6) , 6 6 = 2 − 𝑖√2.

𝑥 3 + 6𝑥 2 + 2𝑥 + 12 = 0 𝑥1 = 𝑦1 − 2 = −6 , 𝑥2 = 𝑦2 − 2 = 𝑖√2 , 𝑥3 = 𝑦3 − 2 = −𝑖√2 .