Solve (a-5)x^2-4x-1=0

$$( a - 5 ) x ^ { 2 } - 4 x - 1 = 0$$

Solve for a

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

$$x\neq 0$$

Steps for Solving Linear Equation

Use the distributive property to multiply a-5 by x^{2}.

$$ax^{2}-5x^{2}-4x-1=0$$

Add 5x^{2} to both sides. Anything plus zero gives itself.

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

Add 4x to both sides.

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

Add 1 to both sides.

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

The equation is in standard form.

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

Divide both sides by x^{2}.

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

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

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

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

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

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{a-1}+2}{a-5}\text{; }x=\frac{-\sqrt{a-1}+2}{a-5}\text{, }&a\neq 5\\x=-\frac{1}{4}\text{, }&a=5\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{a-1}+2}{a-5}\text{; }x=\frac{-\sqrt{a-1}+2}{a-5}\text{, }&a\neq 5\text{ and }a\geq 1\\x=-\frac{1}{4}\text{, }&a=5\end{matrix}\right.$$