Solve x-2/b-2+2/b^2-4b+4+left(1/b-2+x/b-1right)left(b-1right)=0

$$\frac{ x-2 }{ b-2 } + \frac{ 2 }{ { b }^{ 2 } -4b+4 } + \left( \frac{ 1 }{ b-2 } + \frac{ x }{ b-1 } \right) \left( b-1 \right) = 0$$

Solve for x

$$x=-\frac{b^{2}-5b+8}{\left(b-2\right)\left(b-1\right)}$$

$$b\neq 2\text{ and }b\neq 1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(b-1\right)\left(b-2\right)^{2}, the least common multiple of b-2,b^{2}-4b+4,b-1.

$$\left(b-1\right)\left(b-2\right)\left(x-2\right)+\left(b-1\right)\times 2+\left(\frac{1}{b-2}+\frac{x}{b-1}\right)\left(b-1\right)\left(b-1\right)\left(b-2\right)^{2}=0$$

Use the distributive property to multiply b-1 by b-2 and combine like terms.

$$\left(b^{2}-3b+2\right)\left(x-2\right)+\left(b-1\right)\times 2+\left(\frac{1}{b-2}+\frac{x}{b-1}\right)\left(b-1\right)\left(b-1\right)\left(b-2\right)^{2}=0$$

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

$$b^{2}x-2b^{2}-3bx+6b+2x-4+\left(b-1\right)\times 2+\left(\frac{1}{b-2}+\frac{x}{b-1}\right)\left(b-1\right)\left(b-1\right)\left(b-2\right)^{2}=0$$

Use the distributive property to multiply b-1 by 2.

$$b^{2}x-2b^{2}-3bx+6b+2x-4+2b-2+\left(\frac{1}{b-2}+\frac{x}{b-1}\right)\left(b-1\right)\left(b-1\right)\left(b-2\right)^{2}=0$$

Combine 6b and 2b to get 8b.

$$b^{2}x-2b^{2}-3bx+8b+2x-4-2+\left(\frac{1}{b-2}+\frac{x}{b-1}\right)\left(b-1\right)\left(b-1\right)\left(b-2\right)^{2}=0$$

Subtract 2 from -4 to get -6.

$$b^{2}x-2b^{2}-3bx+8b+2x-6+\left(\frac{1}{b-2}+\frac{x}{b-1}\right)\left(b-1\right)\left(b-1\right)\left(b-2\right)^{2}=0$$

Multiply b-1 and b-1 to get \left(b-1\right)^{2}.

$$b^{2}x-2b^{2}-3bx+8b+2x-6+\left(\frac{1}{b-2}+\frac{x}{b-1}\right)\left(b-1\right)^{2}\left(b-2\right)^{2}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b-2 and b-1 is \left(b-2\right)\left(b-1\right). Multiply \frac{1}{b-2} times \frac{b-1}{b-1}. Multiply \frac{x}{b-1} times \frac{b-2}{b-2}.

$$b^{2}x-2b^{2}-3bx+8b+2x-6+\left(\frac{b-1}{\left(b-2\right)\left(b-1\right)}+\frac{x\left(b-2\right)}{\left(b-2\right)\left(b-1\right)}\right)\left(b-1\right)^{2}\left(b-2\right)^{2}=0$$

Since \frac{b-1}{\left(b-2\right)\left(b-1\right)} and \frac{x\left(b-2\right)}{\left(b-2\right)\left(b-1\right)} have the same denominator, add them by adding their numerators.

$$b^{2}x-2b^{2}-3bx+8b+2x-6+\frac{b-1+x\left(b-2\right)}{\left(b-2\right)\left(b-1\right)}\left(b-1\right)^{2}\left(b-2\right)^{2}=0$$

Do the multiplications in b-1+x\left(b-2\right).

$$b^{2}x-2b^{2}-3bx+8b+2x-6+\frac{b-1+xb-2x}{\left(b-2\right)\left(b-1\right)}\left(b-1\right)^{2}\left(b-2\right)^{2}=0$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(b-1\right)^{2}.

$$b^{2}x-2b^{2}-3bx+8b+2x-6+\frac{b-1+xb-2x}{\left(b-2\right)\left(b-1\right)}\left(b^{2}-2b+1\right)\left(b-2\right)^{2}=0$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(b-2\right)^{2}.

$$b^{2}x-2b^{2}-3bx+8b+2x-6+\frac{b-1+xb-2x}{\left(b-2\right)\left(b-1\right)}\left(b^{2}-2b+1\right)\left(b^{2}-4b+4\right)=0$$

Express \frac{b-1+xb-2x}{\left(b-2\right)\left(b-1\right)}\left(b^{2}-2b+1\right) as a single fraction.

$$b^{2}x-2b^{2}-3bx+8b+2x-6+\frac{\left(b-1+xb-2x\right)\left(b^{2}-2b+1\right)}{\left(b-2\right)\left(b-1\right)}\left(b^{2}-4b+4\right)=0$$

Express \frac{\left(b-1+xb-2x\right)\left(b^{2}-2b+1\right)}{\left(b-2\right)\left(b-1\right)}\left(b^{2}-4b+4\right) as a single fraction.

$$b^{2}x-2b^{2}-3bx+8b+2x-6+\frac{\left(b-1+xb-2x\right)\left(b^{2}-2b+1\right)\left(b^{2}-4b+4\right)}{\left(b-2\right)\left(b-1\right)}=0$$

Factor the expressions that are not already factored in \frac{\left(b-1+xb-2x\right)\left(b^{2}-2b+1\right)\left(b^{2}-4b+4\right)}{\left(b-2\right)\left(b-1\right)}.

$$b^{2}x-2b^{2}-3bx+8b+2x-6+\frac{\left(b-2\right)^{2}\left(b-1\right)^{2}\left(bx-2x+b-1\right)}{\left(b-2\right)\left(b-1\right)}=0$$

Cancel out \left(b-2\right)\left(b-1\right) in both numerator and denominator.

$$b^{2}x-2b^{2}-3bx+8b+2x-6+\left(b-2\right)\left(b-1\right)\left(bx-2x+b-1\right)=0$$

Expand the expression.

$$b^{2}x-2b^{2}-3bx+8b+2x-6+xb^{3}-5xb^{2}+8bx-4x+b^{3}-4b^{2}+5b-2=0$$

Combine b^{2}x and -5xb^{2} to get -4b^{2}x.

$$-4b^{2}x-2b^{2}-3bx+8b+2x-6+xb^{3}+8bx-4x+b^{3}-4b^{2}+5b-2=0$$

Combine -3bx and 8bx to get 5bx.

$$-4b^{2}x-2b^{2}+5bx+8b+2x-6+xb^{3}-4x+b^{3}-4b^{2}+5b-2=0$$

Combine 2x and -4x to get -2x.

$$-4b^{2}x-2b^{2}+5bx+8b-2x-6+xb^{3}+b^{3}-4b^{2}+5b-2=0$$

Combine -2b^{2} and -4b^{2} to get -6b^{2}.

$$-4b^{2}x-6b^{2}+5bx+8b-2x-6+xb^{3}+b^{3}+5b-2=0$$

Combine 8b and 5b to get 13b.

$$-4b^{2}x-6b^{2}+5bx+13b-2x-6+xb^{3}+b^{3}-2=0$$

Subtract 2 from -6 to get -8.

$$-4b^{2}x-6b^{2}+5bx+13b-2x-8+xb^{3}+b^{3}=0$$

Add 6b^{2} to both sides. Anything plus zero gives itself.

$$-4b^{2}x+5bx+13b-2x-8+xb^{3}+b^{3}=6b^{2}$$

Subtract 13b from both sides.

$$-4b^{2}x+5bx-2x-8+xb^{3}+b^{3}=6b^{2}-13b$$

Add 8 to both sides.

$$-4b^{2}x+5bx-2x+xb^{3}+b^{3}=6b^{2}-13b+8$$

Subtract b^{3} from both sides.

$$-4b^{2}x+5bx-2x+xb^{3}=6b^{2}-13b+8-b^{3}$$

Combine all terms containing x.

$$\left(-4b^{2}+5b-2+b^{3}\right)x=6b^{2}-13b+8-b^{3}$$

The equation is in standard form.

$$\left(b^{3}-4b^{2}+5b-2\right)x=8-13b+6b^{2}-b^{3}$$

Divide both sides by -4b^{2}+5b-2+b^{3}.

$$\frac{\left(b^{3}-4b^{2}+5b-2\right)x}{b^{3}-4b^{2}+5b-2}=\frac{\left(b-1\right)\left(-b^{2}+5b-8\right)}{b^{3}-4b^{2}+5b-2}$$

Dividing by -4b^{2}+5b-2+b^{3} undoes the multiplication by -4b^{2}+5b-2+b^{3}.

$$x=\frac{\left(b-1\right)\left(-b^{2}+5b-8\right)}{b^{3}-4b^{2}+5b-2}$$

Divide \left(-8+5b-b^{2}\right)\left(-1+b\right) by -4b^{2}+5b-2+b^{3}.

$$x=\frac{-b^{2}+5b-8}{\left(b-2\right)\left(b-1\right)}$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=\frac{\sqrt{x^{2}-10x-7}+3x+5}{2\left(x+1\right)}\text{; }b=\frac{-\sqrt{x^{2}-10x-7}+3x+5}{2\left(x+1\right)}\text{, }&x\neq -1\\b=3\text{, }&x=-1\end{matrix}\right.$$

Solve for b

$$\left\{\begin{matrix}b=\frac{\sqrt{x^{2}-10x-7}+3x+5}{2\left(x+1\right)}\text{; }b=\frac{-\sqrt{x^{2}-10x-7}+3x+5}{2\left(x+1\right)}\text{, }&\left(x\neq -1\text{ and }x\leq 5-4\sqrt{2}\right)\text{ or }x\geq 4\sqrt{2}+5\\b=3\text{, }&x=-1\end{matrix}\right.$$