Solve x=x(x-a)(x-b)

$$x = x ( x - a ) ( x - b )$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{x^{2}-bx-1}{b-x}\text{, }&x\neq b\\a\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$x=\left(x^{2}-xa\right)\left(x-b\right)$$

Use the distributive property to multiply x^{2}-xa by x-b.

$$x=x^{3}-x^{2}b-ax^{2}+axb$$

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

$$x^{3}-x^{2}b-ax^{2}+axb=x$$

Subtract x^{3} from both sides.

$$-x^{2}b-ax^{2}+axb=x-x^{3}$$

Add x^{2}b to both sides.

$$-ax^{2}+axb=x-x^{3}+x^{2}b$$

Combine all terms containing a.

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

The equation is in standard form.

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

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

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

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

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

Divide x\left(1-x^{2}+xb\right) by -x^{2}+xb.

$$a=\frac{1+bx-x^{2}}{b-x}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{x^{2}-ax-1}{a-x}\text{, }&x\neq a\\b\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$x=\left(x^{2}-xa\right)\left(x-b\right)$$

Use the distributive property to multiply x^{2}-xa by x-b.

$$x=x^{3}-x^{2}b-ax^{2}+xba$$

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

$$x^{3}-x^{2}b-ax^{2}+xba=x$$

Subtract x^{3} from both sides.

$$-x^{2}b-ax^{2}+xba=x-x^{3}$$

Add ax^{2} to both sides.

$$-x^{2}b+xba=x-x^{3}+ax^{2}$$

Combine all terms containing b.

$$\left(-x^{2}+xa\right)b=x-x^{3}+ax^{2}$$

The equation is in standard form.

$$\left(ax-x^{2}\right)b=x+ax^{2}-x^{3}$$

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

$$\frac{\left(ax-x^{2}\right)b}{ax-x^{2}}=\frac{x\left(1+ax-x^{2}\right)}{ax-x^{2}}$$

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

$$b=\frac{x\left(1+ax-x^{2}\right)}{ax-x^{2}}$$

Divide x\left(1-x^{2}+ax\right) by -x^{2}+xa.

$$b=\frac{1+ax-x^{2}}{a-x}$$