Solve 3x^2-6x=ax^2-1

$$3 x ^ { 2 } - 6 x = a x ^ { 2 } - 1$$

Solve for a

$$a=3-\frac{6}{x}+\frac{1}{x^{2}}$$

$$x\neq 0$$

Steps for Solving Linear Equation

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

$$ax^{2}-1=3x^{2}-6x$$

Add 1 to both sides.

$$ax^{2}=3x^{2}-6x+1$$

The equation is in standard form.

$$x^{2}a=3x^{2}-6x+1$$

Divide both sides by x^{2}.

$$\frac{x^{2}a}{x^{2}}=\frac{3x^{2}-6x+1}{x^{2}}$$

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

$$a=\frac{3x^{2}-6x+1}{x^{2}}$$

Divide 3x^{2}-6x+1 by x^{2}.

$$a=3-\frac{6}{x}+\frac{1}{x^{2}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{a+6}+3}{3-a}\text{; }x=\frac{-\sqrt{a+6}+3}{3-a}\text{, }&a\neq 3\\x=\frac{1}{6}\text{, }&a=3\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{a+6}+3}{3-a}\text{; }x=\frac{-\sqrt{a+6}+3}{3-a}\text{, }&a\neq 3\text{ and }a\geq -6\\x=\frac{1}{6}\text{, }&a=3\end{matrix}\right.$$