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.