Question

2){2x2 – 3y2 = -19,
{xy = -6;

3){x2+y2=65
{xy=28

Answer 1

2)$\left \{ {{2x^{2}-3y^{2}=-19} \atop {xy=-6}} \right.$

$\left \{ {{2*(-\frac{6}{y} )^{2}-3y^{2}=-19} \atop {x=-\frac{6}{y} }} \right.$

$\left \{ {{2*\frac{36}{y^{2}} -3y^{2}=-19} \atop {x=-\frac{6}{y} }} \right.$

$\left \{ {{\frac{72}{y^{2}} -3y^{2}=-19} \atop {x=-\frac{6}{y} }} \right.$

${\frac{72}{y^{2}} -3y^{2}=-19$ | * y²

72-3y⁴=-19y²

-3y⁴+19y²+72=0 | :(-1)

3y⁴-19y²-72=0

Пусть y²=t, t>0

3t²-19t-72=0

D = (-19)²-4*3*(-72) = 361+864 = 1225 = 35²

$t_{12}=\frac{19б35}{2*3}$

$t_{1} =\frac{54}{6} = 9$

$t_{2}=- \frac{16}{6} = -\frac{8}{3}$ <0

Вернёмся к замене

y² = 9

y = ±3

x₁ = - (6 : 3) = -2

x₂ = - (6 : (-3)) = 2

Ответ: (-2;3),(2;-3)

3)$\left \{ {{x^{2}+y{2}=65} \atop {xy=28}} \right.$

$\left \{ {{(\frac{28}{y} )^{2}+y{2}=65} \atop {x=\frac{28}{y} }} \right.$

$\left \{ {{\frac{784}{y^{2}}+y{2}=65} \atop {x=\frac{28}{y} }} \right.$

${{\frac{784}{y^{2}}+y{2}=65$ | * y²

784 + y⁴ = 65y²

y⁴ - 65y² + 784 = 0

Пусть y²=t, t>0

t²-65t+784=0

D = (-65)²-4*1*784 = 4225 - 3136 = 1089 = 33²

$t_{12}=\frac{65б33}{2*1}$

$t_{1} = \frac{98}{2} = 49$

$t_{2} = \frac{32}{2} = 16$

Вернёмся к замене

$\left \{ {{y^{2}=49} \atop {y^{2}=16}} \right.$

$\left \{ {{y=б7} \atop {y=б4}} \right.$

x₁ = 28:7 = 4

x₂ = 28:(-7) = -4

x₃ = 28:4 = 7

x₄ = 28:(-4) = -7

Ответ: (4;7),(-4;-7),(7;4),(-7;-4)