Solve y=a/x-1-x

$$y = \frac { a } { x } - 1 - x$$

Solve for a

$$a=x\left(x+y+1\right)$$

$$x\neq 0$$

Solution Steps

Multiply both sides of the equation by x.

$$yx=a+x\left(-1\right)-xx$$

Multiply x and x to get x^{2}.

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

Swap sides so that all variable terms are on the left hand side.

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

Subtract x\left(-1\right) from both sides.

$$a-x^{2}=yx-x\left(-1\right)$$

Add x^{2} to both sides.

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

Multiply -1 and -1 to get 1.

$$a=yx+x+x^{2}$$

Solve for x (complex solution)

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

Solve for x

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