Resolver fx^2+y^2-2x-6y+5=0

$$f x ^ { 2 } + y ^ { 2 } - 2 x - 6 y + 5 = 0$$

Ebatzi: f (complex solution)

$$\left\{\begin{matrix}f=-\frac{5-6y+y^{2}-2x}{x^{2}}\text{, }&x\neq 0\\f\in \mathrm{C}\text{, }&\left(y=5\text{ or }y=1\right)\text{ and }x=0\end{matrix}\right.$$

Ekuazio lineala ebazteko urratsak

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

$$fx^{2}-2x-6y+5=-y^{2}$$

Gehitu 2x bi aldeetan.

$$fx^{2}-6y+5=-y^{2}+2x$$

Gehitu 6y bi aldeetan.

$$fx^{2}+5=-y^{2}+2x+6y$$

Kendu 5 bi aldeetatik.

$$fx^{2}=-y^{2}+2x+6y-5$$

Modu arruntean dago ekuazioa.

$$x^{2}f=2x-y^{2}+6y-5$$

Zatitu ekuazioaren bi aldeak x^{2} balioarekin.

$$\frac{x^{2}f}{x^{2}}=\frac{2x-y^{2}+6y-5}{x^{2}}$$

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

$$f=\frac{2x-y^{2}+6y-5}{x^{2}}$$

Ebatzi: f

$$\left\{\begin{matrix}f=-\frac{5-6y+y^{2}-2x}{x^{2}}\text{, }&x\neq 0\\f\in \mathrm{R}\text{, }&\left(y=5\text{ or }y=1\right)\text{ and }x=0\end{matrix}\right.$$

Ekuazio lineala ebazteko urratsak

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

$$fx^{2}-2x-6y+5=-y^{2}$$

Gehitu 2x bi aldeetan.

$$fx^{2}-6y+5=-y^{2}+2x$$

Gehitu 6y bi aldeetan.

$$fx^{2}+5=-y^{2}+2x+6y$$

Kendu 5 bi aldeetatik.

$$fx^{2}=-y^{2}+2x+6y-5$$

Modu arruntean dago ekuazioa.

$$x^{2}f=2x-y^{2}+6y-5$$

Zatitu ekuazioaren bi aldeak x^{2} balioarekin.

$$\frac{x^{2}f}{x^{2}}=\frac{2x-y^{2}+6y-5}{x^{2}}$$

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

$$f=\frac{2x-y^{2}+6y-5}{x^{2}}$$

Ebatzi: x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{1-5f+6fy-fy^{2}}+1}{f}\text{; }x=\frac{-\sqrt{1-5f+6fy-fy^{2}}+1}{f}\text{, }&f\neq 0\\x=\frac{\left(y-5\right)\left(y-1\right)}{2}\text{, }&f=0\end{matrix}\right.$$

Ebatzi: x

$$\left\{\begin{matrix}x=\frac{\sqrt{1-5f+6fy-fy^{2}}+1}{f}\text{; }x=\frac{-\sqrt{1-5f+6fy-fy^{2}}+1}{f}\text{, }&\left(f\neq 0\text{ and }y=1\right)\text{ or }\left(f\neq 0\text{ and }y\leq 1\text{ and }f\leq -\frac{1}{-y^{2}+6y-5}\right)\text{ or }\left(f\neq 0\text{ and }y=5\right)\text{ or }\left(f\neq 0\text{ and }y\geq 5\text{ and }f\leq -\frac{1}{-y^{2}+6y-5}\right)\text{ or }\left(f\neq 0\text{ and }f\geq -\frac{1}{-y^{2}+6y-5}\text{ and }y\geq 1\text{ and }y\leq 5\right)\text{ or }\left(y\neq 1\text{ and }y\neq 5\text{ and }f=-\frac{1}{-y^{2}+6y-5}\right)\\x=\frac{\left(y-5\right)\left(y-1\right)}{2}\text{, }&f=0\end{matrix}\right.$$