Solve A(x^2-x)+B(x^2+x)=4x^2+8x

$$A ( x ^ { 2 } - x ) + B ( x ^ { 2 } + x ) = 4 x ^ { 2 } + 8 x$$

Solve for A (complex solution)

$$\left\{\begin{matrix}A=-\frac{Bx-4x+B-8}{x-1}\text{, }&x\neq 1\\A\in \mathrm{C}\text{, }&x=0\text{ or }\left(B=6\text{ and }x=1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$Ax^{2}-Ax+B\left(x^{2}+x\right)=4x^{2}+8x$$

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

$$Ax^{2}-Ax+Bx^{2}+Bx=4x^{2}+8x$$

Subtract Bx^{2} from both sides.

$$Ax^{2}-Ax+Bx=4x^{2}+8x-Bx^{2}$$

Subtract Bx from both sides.

$$Ax^{2}-Ax=4x^{2}+8x-Bx^{2}-Bx$$

Reorder the terms.

$$Ax^{2}-Ax=-Bx^{2}+4x^{2}-Bx+8x$$

Combine all terms containing A.

$$\left(x^{2}-x\right)A=-Bx^{2}+4x^{2}-Bx+8x$$

The equation is in standard form.

$$\left(x^{2}-x\right)A=8x-Bx+4x^{2}-Bx^{2}$$

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

$$\frac{\left(x^{2}-x\right)A}{x^{2}-x}=\frac{x\left(8-B+4x-Bx\right)}{x^{2}-x}$$

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

$$A=\frac{x\left(8-B+4x-Bx\right)}{x^{2}-x}$$

Divide x\left(-Bx+4x-B+8\right) by x^{2}-x.

$$A=\frac{8-B+4x-Bx}{x-1}$$

Solve for B (complex solution)

$$\left\{\begin{matrix}B=-\frac{Ax-4x-A-8}{x+1}\text{, }&x\neq -1\\B\in \mathrm{C}\text{, }&x=0\text{ or }\left(x=-1\text{ and }A=-2\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$Ax^{2}-Ax+B\left(x^{2}+x\right)=4x^{2}+8x$$

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

$$Ax^{2}-Ax+Bx^{2}+Bx=4x^{2}+8x$$

Subtract Ax^{2} from both sides.

$$-Ax+Bx^{2}+Bx=4x^{2}+8x-Ax^{2}$$

Add Ax to both sides.

$$Bx^{2}+Bx=4x^{2}+8x-Ax^{2}+Ax$$

Reorder the terms.

$$Bx^{2}+Bx=-Ax^{2}+4x^{2}+Ax+8x$$

Combine all terms containing B.

$$\left(x^{2}+x\right)B=-Ax^{2}+4x^{2}+Ax+8x$$

The equation is in standard form.

$$\left(x^{2}+x\right)B=8x+Ax+4x^{2}-Ax^{2}$$

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

$$\frac{\left(x^{2}+x\right)B}{x^{2}+x}=\frac{x\left(8+A+4x-Ax\right)}{x^{2}+x}$$

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

$$B=\frac{x\left(8+A+4x-Ax\right)}{x^{2}+x}$$

Divide x\left(-Ax+4x+A+8\right) by x^{2}+x.

$$B=\frac{8+A+4x-Ax}{x+1}$$

Solve for A

$$\left\{\begin{matrix}A=-\frac{Bx-4x+B-8}{x-1}\text{, }&x\neq 1\\A\in \mathrm{R}\text{, }&x=0\text{ or }\left(B=6\text{ and }x=1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$Ax^{2}-Ax+B\left(x^{2}+x\right)=4x^{2}+8x$$

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

$$Ax^{2}-Ax+Bx^{2}+Bx=4x^{2}+8x$$

Subtract Bx^{2} from both sides.

$$Ax^{2}-Ax+Bx=4x^{2}+8x-Bx^{2}$$

Subtract Bx from both sides.

$$Ax^{2}-Ax=4x^{2}+8x-Bx^{2}-Bx$$

Reorder the terms.

$$Ax^{2}-Ax=-Bx^{2}+4x^{2}-Bx+8x$$

Combine all terms containing A.

$$\left(x^{2}-x\right)A=-Bx^{2}+4x^{2}-Bx+8x$$

The equation is in standard form.

$$\left(x^{2}-x\right)A=8x-Bx+4x^{2}-Bx^{2}$$

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

$$\frac{\left(x^{2}-x\right)A}{x^{2}-x}=\frac{x\left(8-B+4x-Bx\right)}{x^{2}-x}$$

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

$$A=\frac{x\left(8-B+4x-Bx\right)}{x^{2}-x}$$

Divide x\left(-Bx+4x-B+8\right) by x^{2}-x.

$$A=\frac{8-B+4x-Bx}{x-1}$$

Solve for B

$$\left\{\begin{matrix}B=-\frac{Ax-4x-A-8}{x+1}\text{, }&x\neq -1\\B\in \mathrm{R}\text{, }&x=0\text{ or }\left(x=-1\text{ and }A=-2\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$Ax^{2}-Ax+B\left(x^{2}+x\right)=4x^{2}+8x$$

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

$$Ax^{2}-Ax+Bx^{2}+Bx=4x^{2}+8x$$

Subtract Ax^{2} from both sides.

$$-Ax+Bx^{2}+Bx=4x^{2}+8x-Ax^{2}$$

Add Ax to both sides.

$$Bx^{2}+Bx=4x^{2}+8x-Ax^{2}+Ax$$

Reorder the terms.

$$Bx^{2}+Bx=-Ax^{2}+4x^{2}+Ax+8x$$

Combine all terms containing B.

$$\left(x^{2}+x\right)B=-Ax^{2}+4x^{2}+Ax+8x$$

The equation is in standard form.

$$\left(x^{2}+x\right)B=8x+Ax+4x^{2}-Ax^{2}$$

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

$$\frac{\left(x^{2}+x\right)B}{x^{2}+x}=\frac{x\left(8+A+4x-Ax\right)}{x^{2}+x}$$

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

$$B=\frac{x\left(8+A+4x-Ax\right)}{x^{2}+x}$$

Divide x\left(-Ax+4x+A+8\right) by x^{2}+x.

$$B=\frac{8+A+4x-Ax}{x+1}$$