Solve a(x-3)(x-1)=1

$$a ( x - 3 ) ( x - 1 ) = 1$$

Solve for a

$$a=\frac{1}{\left(x-3\right)\left(x-1\right)}$$

$$x\neq 1\text{ and }x\neq 3$$

Steps for Solving Linear Equation

Use the distributive property to multiply a by x-3.

$$\left(ax-3a\right)\left(x-1\right)=1$$

Use the distributive property to multiply ax-3a by x-1 and combine like terms.

$$ax^{2}-4ax+3a=1$$

Combine all terms containing a.

$$\left(x^{2}-4x+3\right)a=1$$

Divide both sides by x^{2}-4x+3.

$$\frac{\left(x^{2}-4x+3\right)a}{x^{2}-4x+3}=\frac{1}{x^{2}-4x+3}$$

Dividing by x^{2}-4x+3 undoes the multiplication by x^{2}-4x+3.

$$a=\frac{1}{x^{2}-4x+3}$$

Divide 1 by x^{2}-4x+3.

$$a=\frac{1}{\left(x-3\right)\left(x-1\right)}$$

Solve for x (complex solution)

$$x=\frac{\sqrt{a^{2}+a}}{a}+2$$

$$x=-\frac{\sqrt{a^{2}+a}}{a}+2\text{, }a\neq 0$$

Solve for x

$$x=\frac{\sqrt{a^{2}+a}}{a}+2$$

$$x=-\frac{\sqrt{a^{2}+a}}{a}+2\text{, }a>0\text{ or }a\leq -1$$