Rešite y=(x-1/x+1)^2w

$$y = ( \frac { x - 1 } { x + 1 } ) ^ { 2 } w$$

Rešitev za w (complex solution)

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

Koraki za reševanje linearnih enačb

Če želite dobiti potenco vrednosti \frac{x-1}{x+1}, potencirajte števec in imenovalec, nato pa delite.

$$y=\frac{\left(x-1\right)^{2}}{\left(x+1\right)^{2}}w$$

Izrazite \frac{\left(x-1\right)^{2}}{\left(x+1\right)^{2}}w kot enojni ulomek.

$$y=\frac{\left(x-1\right)^{2}w}{\left(x+1\right)^{2}}$$

Uporabite binomski izrek \left(a-b\right)^{2}=a^{2}-2ab+b^{2}, da razširite \left(x-1\right)^{2}.

$$y=\frac{\left(x^{2}-2x+1\right)w}{\left(x+1\right)^{2}}$$

Uporabite binomski izrek \left(a+b\right)^{2}=a^{2}+2ab+b^{2}, da razširite \left(x+1\right)^{2}.

$$y=\frac{\left(x^{2}-2x+1\right)w}{x^{2}+2x+1}$$

Zamenjajte strani tako, da so vse spremenljivke na levi strani.

$$\frac{\left(x^{2}-2x+1\right)w}{x^{2}+2x+1}=y$$

Uporabite distributivnost, da pomnožite x^{2}-2x+1 s/z w.

$$\frac{x^{2}w-2xw+w}{x^{2}+2x+1}=y$$

Pomnožite obe strani enačbe s/z \left(x+1\right)^{2}.

$$x^{2}w-2xw+w=y\left(x+1\right)^{2}$$

Uporabite binomski izrek \left(a+b\right)^{2}=a^{2}+2ab+b^{2}, da razširite \left(x+1\right)^{2}.

$$x^{2}w-2xw+w=y\left(x^{2}+2x+1\right)$$

Uporabite distributivnost, da pomnožite y s/z x^{2}+2x+1.

$$x^{2}w-2xw+w=yx^{2}+2yx+y$$

Združite vse člene, ki vsebujejo w.

$$\left(x^{2}-2x+1\right)w=yx^{2}+2yx+y$$

Enačba je v standardni obliki.

$$\left(x^{2}-2x+1\right)w=2xy+yx^{2}+y$$

Delite obe strani z vrednostjo x^{2}-2x+1.

$$\frac{\left(x^{2}-2x+1\right)w}{x^{2}-2x+1}=\frac{y\left(x+1\right)^{2}}{x^{2}-2x+1}$$

Z deljenjem s/z x^{2}-2x+1 razveljavite množenje s/z x^{2}-2x+1.

$$w=\frac{y\left(x+1\right)^{2}}{x^{2}-2x+1}$$

Delite y\left(1+x\right)^{2} s/z x^{2}-2x+1.

$$w=\frac{y\left(x+1\right)^{2}}{\left(x-1\right)^{2}}$$

Rešitev za w

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

Koraki za reševanje linearnih enačb

Če želite dobiti potenco vrednosti \frac{x-1}{x+1}, potencirajte števec in imenovalec, nato pa delite.

$$y=\frac{\left(x-1\right)^{2}}{\left(x+1\right)^{2}}w$$

Izrazite \frac{\left(x-1\right)^{2}}{\left(x+1\right)^{2}}w kot enojni ulomek.

$$y=\frac{\left(x-1\right)^{2}w}{\left(x+1\right)^{2}}$$

Uporabite binomski izrek \left(a-b\right)^{2}=a^{2}-2ab+b^{2}, da razširite \left(x-1\right)^{2}.

$$y=\frac{\left(x^{2}-2x+1\right)w}{\left(x+1\right)^{2}}$$

Uporabite binomski izrek \left(a+b\right)^{2}=a^{2}+2ab+b^{2}, da razširite \left(x+1\right)^{2}.

$$y=\frac{\left(x^{2}-2x+1\right)w}{x^{2}+2x+1}$$

Zamenjajte strani tako, da so vse spremenljivke na levi strani.

$$\frac{\left(x^{2}-2x+1\right)w}{x^{2}+2x+1}=y$$

Uporabite distributivnost, da pomnožite x^{2}-2x+1 s/z w.

$$\frac{x^{2}w-2xw+w}{x^{2}+2x+1}=y$$

Pomnožite obe strani enačbe s/z \left(x+1\right)^{2}.

$$x^{2}w-2xw+w=y\left(x+1\right)^{2}$$

Uporabite binomski izrek \left(a+b\right)^{2}=a^{2}+2ab+b^{2}, da razširite \left(x+1\right)^{2}.

$$x^{2}w-2xw+w=y\left(x^{2}+2x+1\right)$$

Uporabite distributivnost, da pomnožite y s/z x^{2}+2x+1.

$$x^{2}w-2xw+w=yx^{2}+2yx+y$$

Združite vse člene, ki vsebujejo w.

$$\left(x^{2}-2x+1\right)w=yx^{2}+2yx+y$$

Enačba je v standardni obliki.

$$\left(x^{2}-2x+1\right)w=2xy+yx^{2}+y$$

Delite obe strani z vrednostjo x^{2}-2x+1.

$$\frac{\left(x^{2}-2x+1\right)w}{x^{2}-2x+1}=\frac{y\left(x+1\right)^{2}}{x^{2}-2x+1}$$

Z deljenjem s/z x^{2}-2x+1 razveljavite množenje s/z x^{2}-2x+1.

$$w=\frac{y\left(x+1\right)^{2}}{x^{2}-2x+1}$$

Delite y\left(1+x\right)^{2} s/z x^{2}-2x+1.

$$w=\frac{y\left(x+1\right)^{2}}{\left(x-1\right)^{2}}$$

Rešitev za x (complex solution)

$$\left\{\begin{matrix}x=\frac{2\sqrt{wy}+w+y}{w-y}\text{; }x=\frac{-2\sqrt{wy}+w+y}{w-y}\text{, }&w\neq 0\text{ and }y\neq w\\x=0\text{, }&y=w\text{ and }w\neq 0\\x\neq -1\text{, }&y=0\text{ and }w=0\end{matrix}\right,$$

Rešitev za x

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