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

Apr 5, 2020

#1
+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
+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