Find all values of $x$ such that

\frac{x}{x - 5} = \frac{4}{2x - 4} + 2x

LiIIiam0216 Jun 8, 2024

#1**+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! :)

NotThatSmart Jun 8, 2024

#1**+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