Solve 2x^2-1/x^2(x-1)=A/(x-1)+B/x+C/x^2

$$\frac { 2 x ^ { 2 } - 1 } { x ^ { 2 } ( x - 1 ) } = \frac { A } { ( x - 1 ) } + \frac { B } { x } + \frac { C } { x ^ { 2 } }$$

Solve for B (complex solution)

$$B=-\frac{-Ax^{2}+2x^{2}-Cx+C-1}{x\left(1-x\right)}$$

$$x\neq 0\text{ and }x\neq 1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-1\right)x^{2}, the least common multiple of x^{2}\left(x-1\right),x-1,x,x^{2}.

$$2x^{2}-1=x^{2}A+\left(x-1\right)xB+\left(x-1\right)C$$

Use the distributive property to multiply x-1 by x.

$$2x^{2}-1=x^{2}A+\left(x^{2}-x\right)B+\left(x-1\right)C$$

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

$$2x^{2}-1=x^{2}A+x^{2}B-xB+\left(x-1\right)C$$

Use the distributive property to multiply x-1 by C.

$$2x^{2}-1=x^{2}A+x^{2}B-xB+xC-C$$

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

$$x^{2}A+x^{2}B-xB+xC-C=2x^{2}-1$$

Subtract x^{2}A from both sides.

$$x^{2}B-xB+xC-C=2x^{2}-1-x^{2}A$$

Subtract xC from both sides.

$$x^{2}B-xB-C=2x^{2}-1-x^{2}A-xC$$

Add C to both sides.

$$x^{2}B-xB=2x^{2}-1-x^{2}A-xC+C$$

Reorder the terms.

$$Bx^{2}-Bx=-Ax^{2}+2x^{2}-Cx+C-1$$

Combine all terms containing B.

$$\left(x^{2}-x\right)B=-Ax^{2}+2x^{2}-Cx+C-1$$

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

$$\frac{\left(x^{2}-x\right)B}{x^{2}-x}=\frac{-Ax^{2}+2x^{2}-Cx+C-1}{x^{2}-x}$$

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

$$B=\frac{-Ax^{2}+2x^{2}-Cx+C-1}{x^{2}-x}$$

Divide -Ax^{2}+2x^{2}-Cx+C-1 by x^{2}-x.

$$B=\frac{-Ax^{2}+2x^{2}-Cx+C-1}{x\left(x-1\right)}$$

Solve for A

$$A=\frac{-Bx^{2}+2x^{2}+Bx-Cx+C-1}{x^{2}}$$

$$x\neq 0\text{ and }x\neq 1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-1\right)x^{2}, the least common multiple of x^{2}\left(x-1\right),x-1,x,x^{2}.

$$2x^{2}-1=x^{2}A+\left(x-1\right)xB+\left(x-1\right)C$$

Use the distributive property to multiply x-1 by x.

$$2x^{2}-1=x^{2}A+\left(x^{2}-x\right)B+\left(x-1\right)C$$

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

$$2x^{2}-1=x^{2}A+x^{2}B-xB+\left(x-1\right)C$$

Use the distributive property to multiply x-1 by C.

$$2x^{2}-1=x^{2}A+x^{2}B-xB+xC-C$$

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

$$x^{2}A+x^{2}B-xB+xC-C=2x^{2}-1$$

Subtract x^{2}B from both sides.

$$x^{2}A-xB+xC-C=2x^{2}-1-x^{2}B$$

Add xB to both sides.

$$x^{2}A+xC-C=2x^{2}-1-x^{2}B+xB$$

Subtract xC from both sides.

$$x^{2}A-C=2x^{2}-1-x^{2}B+xB-xC$$

Add C to both sides.

$$x^{2}A=2x^{2}-1-x^{2}B+xB-xC+C$$

Reorder the terms.

$$Ax^{2}=-Bx^{2}+2x^{2}+Bx-Cx+C-1$$

The equation is in standard form.

$$x^{2}A=-Bx^{2}+2x^{2}+Bx-Cx+C-1$$

Divide both sides by x^{2}.

$$\frac{x^{2}A}{x^{2}}=\frac{-Bx^{2}+2x^{2}+Bx-Cx+C-1}{x^{2}}$$

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

$$A=\frac{-Bx^{2}+2x^{2}+Bx-Cx+C-1}{x^{2}}$$