Solve (x)=(k-2)*x^3-(k-1)x^2+k*x-6

$$( x ) = ( k - 2 ) \cdot x ^ { 3 } - ( k - 1 ) x ^ { 2 } + k \cdot x - 6$$

Solve for k

$$k=-\frac{-2x^{3}+x^{2}-x-6}{x\left(x^{2}-x+1\right)}$$

$$x\neq 0$$

Steps for Solving Linear Equation

Use the distributive property to multiply k-2 by x^{3}.

$$x=kx^{3}-2x^{3}-\left(k-1\right)x^{2}+kx-6$$

Use the distributive property to multiply k-1 by x^{2}.

$$x=kx^{3}-2x^{3}-\left(kx^{2}-x^{2}\right)+kx-6$$

To find the opposite of kx^{2}-x^{2}, find the opposite of each term.

$$x=kx^{3}-2x^{3}-kx^{2}+x^{2}+kx-6$$

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

$$kx^{3}-2x^{3}-kx^{2}+x^{2}+kx-6=x$$

Add 2x^{3} to both sides.

$$kx^{3}-kx^{2}+x^{2}+kx-6=x+2x^{3}$$

Subtract x^{2} from both sides.

$$kx^{3}-kx^{2}+kx-6=x+2x^{3}-x^{2}$$

Add 6 to both sides.

$$kx^{3}-kx^{2}+kx=x+2x^{3}-x^{2}+6$$

Combine all terms containing k.

$$\left(x^{3}-x^{2}+x\right)k=x+2x^{3}-x^{2}+6$$

The equation is in standard form.

$$\left(x^{3}-x^{2}+x\right)k=2x^{3}-x^{2}+x+6$$

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

$$\frac{\left(x^{3}-x^{2}+x\right)k}{x^{3}-x^{2}+x}=\frac{2x^{3}-x^{2}+x+6}{x^{3}-x^{2}+x}$$

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

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

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

$$k=\frac{2x^{3}-x^{2}+x+6}{x\left(x^{2}-x+1\right)}$$