Solve y=(a-4)x^2+(a-5)x-1

$$y = ( a - 4 ) x ^ { 2 } + ( a - 5 ) x - 1$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{4x^{2}+5x+y+1}{x\left(x+1\right)}\text{, }&x\neq -1\text{ and }x\neq 0\\a\in \mathrm{C}\text{, }&\left(y=-1\text{ and }x=0\right)\text{ or }\left(y=0\text{ and }x=-1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply a-4 by x^{2}.

$$y=ax^{2}-4x^{2}+\left(a-5\right)x-1$$

Use the distributive property to multiply a-5 by x.

$$y=ax^{2}-4x^{2}+ax-5x-1$$

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

$$ax^{2}-4x^{2}+ax-5x-1=y$$

Add 4x^{2} to both sides.

$$ax^{2}+ax-5x-1=y+4x^{2}$$

Add 5x to both sides.

$$ax^{2}+ax-1=y+4x^{2}+5x$$

Add 1 to both sides.

$$ax^{2}+ax=y+4x^{2}+5x+1$$

Combine all terms containing a.

$$\left(x^{2}+x\right)a=y+4x^{2}+5x+1$$

The equation is in standard form.

$$\left(x^{2}+x\right)a=4x^{2}+5x+y+1$$

Divide both sides by x^{2}+x.

$$\frac{\left(x^{2}+x\right)a}{x^{2}+x}=\frac{4x^{2}+5x+y+1}{x^{2}+x}$$

Dividing by x^{2}+x undoes the multiplication by x^{2}+x.

$$a=\frac{4x^{2}+5x+y+1}{x^{2}+x}$$

Divide y+4x^{2}+1+5x by x^{2}+x.

$$a=\frac{4x^{2}+5x+y+1}{x\left(x+1\right)}$$

Solve for a

$$\left\{\begin{matrix}a=\frac{4x^{2}+5x+y+1}{x\left(x+1\right)}\text{, }&x\neq -1\text{ and }x\neq 0\\a\in \mathrm{R}\text{, }&\left(y=-1\text{ and }x=0\right)\text{ or }\left(y=0\text{ and }x=-1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply a-4 by x^{2}.

$$y=ax^{2}-4x^{2}+\left(a-5\right)x-1$$

Use the distributive property to multiply a-5 by x.

$$y=ax^{2}-4x^{2}+ax-5x-1$$

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

$$ax^{2}-4x^{2}+ax-5x-1=y$$

Add 4x^{2} to both sides.

$$ax^{2}+ax-5x-1=y+4x^{2}$$

Add 5x to both sides.

$$ax^{2}+ax-1=y+4x^{2}+5x$$

Add 1 to both sides.

$$ax^{2}+ax=y+4x^{2}+5x+1$$

Combine all terms containing a.

$$\left(x^{2}+x\right)a=y+4x^{2}+5x+1$$

The equation is in standard form.

$$\left(x^{2}+x\right)a=4x^{2}+5x+y+1$$

Divide both sides by x^{2}+x.

$$\frac{\left(x^{2}+x\right)a}{x^{2}+x}=\frac{4x^{2}+5x+y+1}{x^{2}+x}$$

Dividing by x^{2}+x undoes the multiplication by x^{2}+x.

$$a=\frac{4x^{2}+5x+y+1}{x^{2}+x}$$

Divide y+4x^{2}+1+5x by x^{2}+x.

$$a=\frac{4x^{2}+5x+y+1}{x\left(x+1\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{4ay-16y+a^{2}-6a+9}-a+5}{2\left(a-4\right)}\text{; }x=\frac{-\sqrt{4ay-16y+a^{2}-6a+9}-a+5}{2\left(a-4\right)}\text{, }&a\neq 4\\x=-y-1\text{, }&a=4\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{4ay-16y+a^{2}-6a+9}-a+5}{2\left(a-4\right)}\text{; }x=\frac{-\sqrt{4ay-16y+a^{2}-6a+9}-a+5}{2\left(a-4\right)}\text{, }&\left(a>4\text{ or }y\leq -\frac{\left(a-3\right)^{2}}{4a-16}\right)\text{ and }\left(y\leq \text{Indeterminate}\text{ or }a\neq 4\right)\text{ and }\left(a<4\text{ or }\left(a\neq 4\text{ and }y\geq -\frac{\left(a-3\right)^{2}}{4a-16}\right)\right)\\x=-y-1\text{, }&a=4\end{matrix}\right.$$