Simplify the expression and solve the equation.  `1/(x-2)` +`(2x)/((x-2)(x-8))` =`x/(2(x-8))`   `((m^2+2m+1)/(m^3+3m^2+3m+1))(m^2/m-(3m)/3)` Thank you in advance.

embizze | High School Teacher | (Level 2) Educator Emeritus

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(1) To solve the equation we multiply through by the least common denominator -- 2(x-2)(x-8)

`2(x-2)(x-8)(1/(x-2)+(2x)/((x-2)(x-8)))=2(x-2)(x-8)(x/(2(x-8)))` `2(x-8)+2(2x)=(x-2)x`

`2x-16+4x=x^2-2x`

`x^2-8x+16=0`

`(x-4)^2=0`

x=4

Check: `1/(4-2)+(2(4))/((4-2)(4-8))=1/2+8/(-8)=-1/2=4/(2(4-8))`

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The solution is x=4

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The graph of the left side in black; the right side in red:

(2) Simplify `((m^2+2m+1)/(m^3+3m^2+3m+1))(m^2/m-(3m)/3)`

** Note that the experssion in the second parantheses simplifies:

`m^2/m-(3m)/3=m-m=0` Multiplying a real number by zero always results in zero thus the entire expression simplifies to zero.

** The expression in the first paranthese simplifies -- not that it matters:

`(m^2+2m+1)/(m^3+3m^2+3m+1)=((m+1)^2)/((m+1)^3)=1/(m+1)`