equations: x / x - 2 + x + 1 / X = 8 / x2 - 2x

[dkarolasz]

x x
__ -___= 1
6 8

last one

x x+1 8
___+____=_____
x-2 X x2-2x

I believe the next step is

x x+1 8
____+_____=_______
x+2 x x(x-2)

 

[soroban]

Re: equations

Hello, dkarolasz!

\(\displaystyle \L\frac{x}{6}\,-\,\frac{x}{8}\;=\;1\)

Multiply through by the common denominator: 24.

. . \(\displaystyle \L24\cdot\frac{x}{6}\,-\,24\cdot\frac{x}{8}\;=\;24\cdot1\)

And we have: \(\displaystyle \L\:4x\,-\,3x\:=\:24\;\;\Rightarrow\;\;x\:=\:24\)

\(\displaystyle \L\frac{x}{x\,-\,2}\,+\,\frac{x\,+\,1}{x}\;=\;\frac{8}{x(x\,-\,2)}\)

Multiply through by the common denominator: \(\displaystyle x(x\,-\,2)\)

\(\displaystyle \L x(x\,-\,2)\cdot\frac{x}{x\,-\,2} \,+\,x(x\,-\,2)\cdot\frac{x\,+\,1}{x}\;=\;x(x\,-\,2)\cdot\frac{8}{x(x\,-\,2)}\)

Reduce: \(\displaystyle \L\:x\cdot x \,+\,(x\,-\,2)(x\,+\,1) \;=\;8\)

and we have a quadratic: \(\displaystyle \L\:2x^2\,-\,x\,-\,10\;=\;0\)

. . which factors: \(\displaystyle \L\x\,+\,2)(2x\,-\,5)\;=\;0\)

. . and has roots: \(\displaystyle \L\:x\:=\:-2,\:\frac{5}{2}\)


Are you getting the idea by now?

 

[dkarolasz]

Thank you so much, This is the first time I used this website and its been great, a big help. i understood everything that you wrote and tried to remember all the steps that I might not have understood.

Thanks again.