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Find all values of x that make the equation true:

3/x - 4 = x - 5/x. List all values separated by commas.

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Final answer:

To solve the equation 3/x - 4 = x - 5/x, combine the like terms, create a quadratic equation, and use the quadratic formula to find that the solutions are x = 2 + √2 and x = 2 - √2.

Step-by-step explanation:

To find all values of x that make the equation 3/x - 4 = x - 5/x true, we first combine the terms with x in the denominator:

3/x - 5/x = x - 4

Then, we combine the fractions on the left side of the equation:

(3 - 5) / x = x - 4

Now, we simplify the numerator:

-2/x = x - 4

To solve for x, we convert this to a quadratic equation by multiplying both sides of the equation by x (remembering that x cannot be zero):

-2 = x^2 - 4x

Moving all terms to one side gives us:

x^2 - 4x + 2 = 0

Now we can use the quadratic equation to solve for x, resulting in two potential solutions for x. Subtracting square roots will yield one positive and one negative result:

x = (4 ± √(16-4(1)(2))) / 2(1)

x = (4 ± √(8)) / 2

Let's simplify further:

x = (4 ± 2√2) / 2

x = 2 ± √2

So, the two possible solutions are x = 2 + √2 and x = 2 - √2. We can then use a calculator to find approximate values.

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