Resolver Ex^2+y^2-2x+4y=0

$$E x ^ { 2 } + y ^ { 2 } - 2 x + 4 y = 0$$

Ebatzi: E (complex solution)

$$\left\{\begin{matrix}E=-\frac{4y+y^{2}-2x}{x^{2}}\text{, }&x\neq 0\\E\in \mathrm{C}\text{, }&\left(y=0\text{ or }y=-4\right)\text{ and }x=0\end{matrix}\right.$$

Ekuazio lineala ebazteko urratsak

Kendu y^{2} bi aldeetatik. Zero ken edozein zenbaki zenbaki horren negatiboa da.

$$Ex^{2}-2x+4y=-y^{2}$$

Gehitu 2x bi aldeetan.

$$Ex^{2}+4y=-y^{2}+2x$$

Kendu 4y bi aldeetatik.

$$Ex^{2}=-y^{2}+2x-4y$$

Modu arruntean dago ekuazioa.

$$x^{2}E=2x-y^{2}-4y$$

Zatitu ekuazioaren bi aldeak x^{2} balioarekin.

$$\frac{x^{2}E}{x^{2}}=\frac{2x-y^{2}-4y}{x^{2}}$$

x^{2} balioarekin zatituz gero, x^{2} balioarekiko biderketa desegiten da.

$$E=\frac{2x-y^{2}-4y}{x^{2}}$$

Ebatzi: E

$$\left\{\begin{matrix}E=-\frac{4y+y^{2}-2x}{x^{2}}\text{, }&x\neq 0\\E\in \mathrm{R}\text{, }&\left(y=0\text{ or }y=-4\right)\text{ and }x=0\end{matrix}\right.$$

Ekuazio lineala ebazteko urratsak

Kendu y^{2} bi aldeetatik. Zero ken edozein zenbaki zenbaki horren negatiboa da.

$$Ex^{2}-2x+4y=-y^{2}$$

Gehitu 2x bi aldeetan.

$$Ex^{2}+4y=-y^{2}+2x$$

Kendu 4y bi aldeetatik.

$$Ex^{2}=-y^{2}+2x-4y$$

Modu arruntean dago ekuazioa.

$$x^{2}E=2x-y^{2}-4y$$

Zatitu ekuazioaren bi aldeak x^{2} balioarekin.

$$\frac{x^{2}E}{x^{2}}=\frac{2x-y^{2}-4y}{x^{2}}$$

x^{2} balioarekin zatituz gero, x^{2} balioarekiko biderketa desegiten da.

$$E=\frac{2x-y^{2}-4y}{x^{2}}$$

Ebatzi: x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{1-4Ey-Ey^{2}}+1}{E}\text{; }x=\frac{-\sqrt{1-4Ey-Ey^{2}}+1}{E}\text{, }&E\neq 0\\x=\frac{y\left(y+4\right)}{2}\text{, }&E=0\end{matrix}\right.$$

Ebatzi: x

$$\left\{\begin{matrix}x=\frac{\sqrt{1-4Ey-Ey^{2}}+1}{E}\text{; }x=\frac{-\sqrt{1-4Ey-Ey^{2}}+1}{E}\text{, }&\left(y\neq -4\text{ and }y\neq 0\text{ and }E=\frac{1}{y^{2}+4y}\right)\text{ or }\left(E\neq 0\text{ and }y\geq -4\text{ and }y\leq 0\text{ and }E\geq \frac{1}{y^{2}+4y}\right)\text{ or }\left(E\neq 0\text{ and }y\geq 0\text{ and }E\leq \frac{1}{y^{2}+4y}\right)\text{ or }\left(E\neq 0\text{ and }y=0\right)\text{ or }\left(E\neq 0\text{ and }y\leq -4\text{ and }E\leq \frac{1}{y^{2}+4y}\right)\text{ or }\left(E\neq 0\text{ and }y=-4\right)\text{ or }\left(E=\frac{1}{y^{2}+4y}\text{ and }y<0\text{ and }y>-4\right)\text{ or }\left(E=\frac{1}{y^{2}+4y}\text{ and }y>0\right)\text{ or }\left(E=\frac{1}{y^{2}+4y}\text{ and }y<-4\right)\\x=\frac{y\left(y+4\right)}{2}\text{, }&E=0\end{matrix}\right.$$