Solve ax^2+ax/bx+x^2-2x-3/2x-6=4

$$\frac { a x ^ { 2 } + a x } { b x } + \frac { x ^ { 2 } - 2 x - 3 } { 2 x - 6 } = 4$$

Solve for a

$$a=-\frac{b\left(x-7\right)}{2\left(x+1\right)}$$

$$x\neq -1\text{ and }b\neq 0\text{ and }x\neq 0\text{ and }x\neq 3$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2bx\left(x-3\right), the least common multiple of bx,2x-6.

$$\left(2x-6\right)\left(ax^{2}+ax\right)+bx\left(x^{2}-2x-3\right)=8bx\left(x-3\right)$$

Use the distributive property to multiply 2x-6 by ax^{2}+ax and combine like terms.

$$2ax^{3}-4ax^{2}-6ax+bx\left(x^{2}-2x-3\right)=8bx\left(x-3\right)$$

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

$$2ax^{3}-4ax^{2}-6ax+bx^{3}-2bx^{2}-3bx=8bx\left(x-3\right)$$

Use the distributive property to multiply 8bx by x-3.

$$2ax^{3}-4ax^{2}-6ax+bx^{3}-2bx^{2}-3bx=8bx^{2}-24bx$$

Subtract bx^{3} from both sides.

$$2ax^{3}-4ax^{2}-6ax-2bx^{2}-3bx=8bx^{2}-24bx-bx^{3}$$

Add 2bx^{2} to both sides.

$$2ax^{3}-4ax^{2}-6ax-3bx=8bx^{2}-24bx-bx^{3}+2bx^{2}$$

Add 3bx to both sides.

$$2ax^{3}-4ax^{2}-6ax=8bx^{2}-24bx-bx^{3}+2bx^{2}+3bx$$

Combine 8bx^{2} and 2bx^{2} to get 10bx^{2}.

$$2ax^{3}-4ax^{2}-6ax=10bx^{2}-24bx-bx^{3}+3bx$$

Combine -24bx and 3bx to get -21bx.

$$2ax^{3}-4ax^{2}-6ax=10bx^{2}-21bx-bx^{3}$$

Combine all terms containing a.

$$\left(2x^{3}-4x^{2}-6x\right)a=10bx^{2}-21bx-bx^{3}$$

The equation is in standard form.

$$\left(2x^{3}-4x^{2}-6x\right)a=-bx^{3}+10bx^{2}-21bx$$

Divide both sides by 2x^{3}-4x^{2}-6x.

$$\frac{\left(2x^{3}-4x^{2}-6x\right)a}{2x^{3}-4x^{2}-6x}=\frac{bx\left(3-x\right)\left(x-7\right)}{2x^{3}-4x^{2}-6x}$$

Dividing by 2x^{3}-4x^{2}-6x undoes the multiplication by 2x^{3}-4x^{2}-6x.

$$a=\frac{bx\left(3-x\right)\left(x-7\right)}{2x^{3}-4x^{2}-6x}$$

Divide xb\left(-7+x\right)\left(3-x\right) by 2x^{3}-4x^{2}-6x.

$$a=-\frac{b\left(x-7\right)}{2\left(x+1\right)}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{2a\left(x+1\right)}{x-7}\text{, }&x\neq -1\text{ and }a\neq 0\text{ and }x\neq 7\text{ and }x\neq 0\text{ and }x\neq 3\\b\neq 0\text{, }&x=7\text{ and }a=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable b cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2bx\left(x-3\right), the least common multiple of bx,2x-6.

$$\left(2x-6\right)\left(ax^{2}+ax\right)+xb\left(x^{2}-2x-3\right)=8bx\left(x-3\right)$$

Use the distributive property to multiply 2x-6 by ax^{2}+ax and combine like terms.

$$2ax^{3}-4ax^{2}-6ax+xb\left(x^{2}-2x-3\right)=8bx\left(x-3\right)$$

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

$$2ax^{3}-4ax^{2}-6ax+bx^{3}-2bx^{2}-3xb=8bx\left(x-3\right)$$

Use the distributive property to multiply 8bx by x-3.

$$2ax^{3}-4ax^{2}-6ax+bx^{3}-2bx^{2}-3xb=8bx^{2}-24xb$$

Subtract 8bx^{2} from both sides.

$$2ax^{3}-4ax^{2}-6ax+bx^{3}-2bx^{2}-3xb-8bx^{2}=-24xb$$

Combine -2bx^{2} and -8bx^{2} to get -10bx^{2}.

$$2ax^{3}-4ax^{2}-6ax+bx^{3}-10bx^{2}-3xb=-24xb$$

Add 24xb to both sides.

$$2ax^{3}-4ax^{2}-6ax+bx^{3}-10bx^{2}-3xb+24xb=0$$

Combine -3xb and 24xb to get 21xb.

$$2ax^{3}-4ax^{2}-6ax+bx^{3}-10bx^{2}+21xb=0$$

Subtract 2ax^{3} from both sides. Anything subtracted from zero gives its negation.

$$-4ax^{2}-6ax+bx^{3}-10bx^{2}+21xb=-2ax^{3}$$

Add 4ax^{2} to both sides.

$$-6ax+bx^{3}-10bx^{2}+21xb=-2ax^{3}+4ax^{2}$$

Add 6ax to both sides.

$$bx^{3}-10bx^{2}+21xb=-2ax^{3}+4ax^{2}+6ax$$

Combine all terms containing b.

$$\left(x^{3}-10x^{2}+21x\right)b=-2ax^{3}+4ax^{2}+6ax$$

The equation is in standard form.

$$\left(x^{3}-10x^{2}+21x\right)b=6ax+4ax^{2}-2ax^{3}$$

Divide both sides by x^{3}-10x^{2}+21x.

$$\frac{\left(x^{3}-10x^{2}+21x\right)b}{x^{3}-10x^{2}+21x}=-\frac{2ax\left(x-3\right)\left(x+1\right)}{x^{3}-10x^{2}+21x}$$

Dividing by x^{3}-10x^{2}+21x undoes the multiplication by x^{3}-10x^{2}+21x.

$$b=-\frac{2ax\left(x-3\right)\left(x+1\right)}{x^{3}-10x^{2}+21x}$$

Divide -2ax\left(-3+x\right)\left(1+x\right) by x^{3}-10x^{2}+21x.

$$b=-\frac{2a\left(x+1\right)}{x-7}$$

Variable b cannot be equal to 0.

$$b=-\frac{2a\left(x+1\right)}{x-7}\text{, }b\neq 0$$