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Find all values of $x$ such that
\frac{x}{x - 5} = \frac{4}{2x - 4} + 2x

 Jun 8, 2024

Best Answer 

 #1
avatar+759 
+1

First, we want to get rid of the deniminators. We can do this by multiplying both sides by the LCM of the denominators. 

 

The LCM is (x-5)(x-2). 

 

We have

\(x\left(x-2\right)=2\left(x-5\right)+2x\left(x-5\right)\left(x-2\right)\)

 

We now get a quadratic. 

 

We get 

\(x^2-2x=2x^3-14x^2+22x-10\\ 2x^3-15x^2+24x-10=0\)

 

This where it gets tricky. There are no integer roots for this equation. 

We could use the Newton-Raphson equation, and we eventually get

\(x\approx \:0.67798\dots ,\:x\approx \:1.34697\dots ,\:x\approx \:5.47503\dots \)

 

I hope this was clear enough!

 

Thanks! :)

 Jun 8, 2024
 #1
avatar+759 
+1
Best Answer

First, we want to get rid of the deniminators. We can do this by multiplying both sides by the LCM of the denominators. 

 

The LCM is (x-5)(x-2). 

 

We have

\(x\left(x-2\right)=2\left(x-5\right)+2x\left(x-5\right)\left(x-2\right)\)

 

We now get a quadratic. 

 

We get 

\(x^2-2x=2x^3-14x^2+22x-10\\ 2x^3-15x^2+24x-10=0\)

 

This where it gets tricky. There are no integer roots for this equation. 

We could use the Newton-Raphson equation, and we eventually get

\(x\approx \:0.67798\dots ,\:x\approx \:1.34697\dots ,\:x\approx \:5.47503\dots \)

 

I hope this was clear enough!

 

Thanks! :)

NotThatSmart Jun 8, 2024

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