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Task/26155351
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решите уравнение
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(N 324497) <span>(x² -25)² + (x² +3x -10 ) ² = 0 .
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решение :
(x² -25)² + (x² +3x -10 ) ² = 0 ⇔(равносильно системе) { x² -25 =0 ; <span>
{</span>x² +3x -10 =0 . ⇔<span>
{ [ x = -5</span> ; x= 5 ;
{ [ x = -5 ; x = 3 ⇒ x = -5 .
ответ: x = - 5 .
* * * * * * * * * * * *
(N 324494)
x⁶ = (x- 5)³ ; * * * x² = x -5 * * *
(x²)³ -(x -5)³ = 0 ;
( x² -(x -5) ) *( (x²)² +x²(x -5) +(x-5)²<span> ) =0 ;
</span>x²<span> -(x -5) =0 ;
</span>x² -x +5 =0 D = 1<span>² -4*1*5 = -19 < 0</span> не имеет действительные решения
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(x²)² +x²(x -5) +(x-5)² =0 тоже не имеет действ. <span>реш.
</span>(x²)² +x²(x -5) +(x-5)² = ( ( x²/2)² + (x -5) ) ² +3x⁴ /4 ≠ 0
ответ: x ∈ ∅ .
и что это что с этим надо делоть
![\begin{cases} & \text{ } x^2+xy+x=10 \\ & \text{ } y^2+xy+y=20 \end{cases}\Rightarrow\begin{cases} & \text{ } x(x+y+1)=10 \\ & \text{ } y(x+y+1)=20 \end{cases}\Rightarrow\\ \Rightarrow\begin{cases} & \text{ } 2x(x+y+1-y(x+y+1)=2\cdot10-1\cdot20 \\ & \text{ } y(x+y+1)=20 \end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D%0A%26+%5Ctext%7B+%7D+x%5E2%2Bxy%2Bx%3D10+%5C%5C+%0A%26+%5Ctext%7B+%7D+y%5E2%2Bxy%2By%3D20%0A%5Cend%7Bcases%7D%5CRightarrow%5Cbegin%7Bcases%7D%0A%26+%5Ctext%7B+%7D+x%28x%2By%2B1%29%3D10+%5C%5C+%0A%26+%5Ctext%7B+%7D+y%28x%2By%2B1%29%3D20%0A%5Cend%7Bcases%7D%5CRightarrow%5C%5C+%5CRightarrow%5Cbegin%7Bcases%7D%0A%26+%5Ctext%7B+%7D+2x%28x%2By%2B1-y%28x%2By%2B1%29%3D2%5Ccdot10-1%5Ccdot20+%5C%5C+%0A%26+%5Ctext%7B+%7D+y%28x%2By%2B1%29%3D20+%0A%5Cend%7Bcases%7D)
Система эквивалентна предыдущей. так как
![x+y+1\ne 0](https://tex.z-dn.net/?f=x%2By%2B1%5Cne++0)
, то
![\begin{cases} & \text{ } 2x-y=0 \\ & \text{ } y^2+xy+y=20 \end{cases}\Rightarrow\begin{cases} & \text{ } y=2x \\ & \text{ } (2x)^2+2x\cdot x+2x=20 \end{cases}\\ 4x^2+2x^2+2x=20\\ 6x^2+2x-20=0|:2\\ 3x^2+x-10=0](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D%0A%26+%5Ctext%7B+%7D+2x-y%3D0+%5C%5C+%0A%26+%5Ctext%7B+%7D+y%5E2%2Bxy%2By%3D20+%0A%5Cend%7Bcases%7D%5CRightarrow%5Cbegin%7Bcases%7D%0A%26+%5Ctext%7B+%7D+y%3D2x+%5C%5C+%0A%26+%5Ctext%7B+%7D+%282x%29%5E2%2B2x%5Ccdot+x%2B2x%3D20%0A%5Cend%7Bcases%7D%5C%5C+4x%5E2%2B2x%5E2%2B2x%3D20%5C%5C+6x%5E2%2B2x-20%3D0%7C%3A2%5C%5C+3x%5E2%2Bx-10%3D0)
Находим дискриминант
![D=b^2-4ac=1^2-4\cdot3\cdot (-10)=121;\\ x_1= \frac{-1+11}{6}= \frac{5}{3} \\ x_2= \frac{-1-11}{6}=-2](https://tex.z-dn.net/?f=D%3Db%5E2-4ac%3D1%5E2-4%5Ccdot3%5Ccdot+%28-10%29%3D121%3B%5C%5C+x_1%3D+%5Cfrac%7B-1%2B11%7D%7B6%7D%3D++%5Cfrac%7B5%7D%7B3%7D++%5C%5C+x_2%3D+%5Cfrac%7B-1-11%7D%7B6%7D%3D-2+)
Нашли значение системы х1 и х2, теперь найдем у1 и у2
![y_1=2x_1=2\cdot \frac{5}{3} = \frac{10}{3} \\ y_2=2x_2=2\cdot(-2)=-4](https://tex.z-dn.net/?f=y_1%3D2x_1%3D2%5Ccdot+%5Cfrac%7B5%7D%7B3%7D+%3D+%5Cfrac%7B10%7D%7B3%7D++%5C%5C+y_2%3D2x_2%3D2%5Ccdot%28-2%29%3D-4)
Ответ:
![(-2;-4),\,\,(\frac{5}{3} ;\frac{10}{3} ).](https://tex.z-dn.net/?f=%28-2%3B-4%29%2C%5C%2C%5C%2C%28%5Cfrac%7B5%7D%7B3%7D+%3B%5Cfrac%7B10%7D%7B3%7D+%29.)