let K=1/(1/4) if x=K/(K-x), find x
To find the value of x, we'll substitute the given value K=1/(1/4) into the equation x=K/(K-x) and solve for x.
Let's start by simplifying the given value of K. We have:
K = 1 / (1/4)
K = 1 / (4/1) (reciprocal of 1/4)
K = 1 * (1/4)
K = 1/4
Now, let's substitute this value of K into the equation x = K / (K - x) and solve for x:
x = (1/4) / ((1/4) - x)
Next, we multiply both the numerator and denominator by the least common multiple (LCM) of the fractions involved, which in this case is 4. This step allows us to eliminate the fractions:
4x = 4 * (1/4) / ((1/4) - x)
4x = 1 / (1 - 4x)
To get rid of the fraction in the denominator, we multiply both sides of the equation by (1 - 4x):
4x * (1 - 4x) = 1 * (1 - 4x)
4x - 16x^2 = 1 - 4x
Combining like terms, we get:
-16x^2 = 1 - 4x - 4x
-16x^2 = 1 - 8x
Moving all the terms to one side of the equation, we have:
16x^2 - 8x - 1 = 0
Now, we can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 16, b = -8, and c = -1. Substituting these values into the quadratic formula, we get:
x = (-(−8) ± √((−8)^2 − 4 × 16 × (−1))) / (2 × 16)
x = (8 ± √(64 + 64)) / 32
x = (8 ± √128) / 32
x = (8 ± 8√2) / 32
x = (1 ± √2) / 4
Therefore, we have two possible values for x:
x = (1 + √2) / 4
x = (1 - √2) / 4