To solve for x, we need to first combine the fractions:
4/6 + 1/x = x-4
Next, find a common denominator for the fractions, which in this case would be 6x:
(4x + 6) / 6x + 1/x = x - 4
Now, multiply both sides of the equation by 6x to eliminate the fractions:
4x + 6 + 6 = 6x(x - 4)
Simplify:
4x + 12 = 6x^2 - 24x
Rearrange the equation into standard form:
6x^2 - 24x - 4x - 12 = 0
6x^2 - 28x - 12 = 0
Now, we can solve for x using the quadratic formula:
x = (-(-28) ± √((-28)^2 - 4(6)(-12))) / 2(6)
x = (28 ± √(784 + 288)) / 12
x = (28 ± √1072) / 12
x = (28 ± √(16*67)) / 12
x = (28 ± 4√67) / 12
x = (7 ± √67) / 3
So, the solutions for x are:
x = (7 + √67) / 3 or x = (7 - √67) / 3
To solve for x, we need to first combine the fractions:
4/6 + 1/x = x-4
Next, find a common denominator for the fractions, which in this case would be 6x:
(4x + 6) / 6x + 1/x = x - 4
Now, multiply both sides of the equation by 6x to eliminate the fractions:
4x + 6 + 6 = 6x(x - 4)
Simplify:
4x + 12 = 6x^2 - 24x
Rearrange the equation into standard form:
6x^2 - 24x - 4x - 12 = 0
6x^2 - 28x - 12 = 0
Now, we can solve for x using the quadratic formula:
x = (-(-28) ± √((-28)^2 - 4(6)(-12))) / 2(6)
x = (28 ± √(784 + 288)) / 12
x = (28 ± √1072) / 12
x = (28 ± √(16*67)) / 12
x = (28 ± 4√67) / 12
x = (7 ± √67) / 3
So, the solutions for x are:
x = (7 + √67) / 3 or x = (7 - √67) / 3