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Find all values of x such that x/x - 5 = 4/x - 4

 Apr 5, 2020
 #1
avatar+34 
+2

\( \frac{x}{x}-5=\frac{4}{x}-4\)

We know that \(\frac{x}{x}\) is \(1\), therefore meaning that the left side of the equation is, 

\(1-5=-4\)

Therefore our equation is now

 \(-4=\frac{4}{x}-4\)

From there we can simplify by adding 4 to each side 

\(0=\frac{4}{x}\)

There is no value of \(x\) that satisfies this equation. 

Therefore there are no solutions 

 Apr 5, 2020
 #2
avatar+483 
+2

As it was written by the Guest, ako correctly points out that this equation has no solutions. However, I will take it that they forgot parentheses(important!) around the denominator and numerator. I'll then solve it with this interpretation:

 

\({x\over x-5} = {4 \over x-4}\)

Cross multiplying ,we get

\(4x-20 = x^2-4x\)

Rearranging, we get:

\(x^2-8x + 20 = 0\)

Realizing that our roots are imaginary because of the discriminant(but the problem never limits the value of x, so I'll continue), we can use the quadratic formula to find:

 

\(x = {8 \pm \sqrt{-16} \over 2}\)

\(x = {8 \pm 4i \over 2} = 4\pm2i\)

.
 Apr 5, 2020

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