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

 Apr 2, 2020
 #1
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Using Cross Multiplication and Quadratic Equation, I got 4 + 2i and 4 -2i

 Apr 2, 2020
 #2
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I'm going to assume you mean x / ( x - 5 ) = 4 / ( x - 4 ). 
From that equation, we can cross multiply and get x ( x - 4 ) = 4 ( x - 5 ).

Expand. x^2 - 4x = 4x - 20. 
To make it easier to find all values of x, set the equation equal to zero. x^2 - 8x + 20 = 0. 
Plugging this into the Quadratic Formula, we get x = [ 8 ± sqrt ( 64 - 4 * 1 * 20 ) ] / ( 2 * 1 ). 
Simplify. x = 4 ± 2i. 

(Simplifiying steps: 

1. Simplify the inside of the square root: x = [ 8 ± sqrt ( -16 ) ] / ( 2 * 1 )

2. Simplify the denominator: x = [ 8 ± sqrt ( -16 ) ] / 2

3. Separate the fraction two parts: x = ( 8 / 2 ) ± [ ( sqrt -16 ) / 2 ]

4. Simplify the two fractions: x = 4 ± 2i

    (Basically doing 8 / 2, and solving the imaginary fraction ( ( i * sqrt 16 ) / 2 = 4i / 2 = 2i ) )
 

I hope I helped! A bit rusty on this topic, so if I made any mistake please lmk!

 Apr 2, 2020
edited by milkcloud  Apr 3, 2020

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