x-14/x-1=4-2x/x-1
Multiply both sides by x.
x^2 - 14 - x = 4x - 2x - x
Combine like terms, set equal to zero and factor the quadratic equation.
To solve this equation, we'll begin by getting rid of the fractions. We can do this by cross-multiplying.
First, let's cross-multiply the first fraction, (x-14)/(x-1), with the right-hand side fraction, (4-2x)/(x-1):
(x-14)(x-1) = (4-2x)(x-1)
Expanding the expressions:
(x^2 - 15x + 14) = (4x - 4 - 2x^2 + 2x)
Next, gather all like terms on one side of the equation:
x^2 - 15x + 14 = 4x - 4 - 2x^2 + 2x
Rearrange the terms by combining like terms:
x^2 - 15x + 14 + 2x^2 - 4x - 2x = 4 - 4
Simplify the equation:
3x^2 - 17x + 14 = 0
Now, we have a quadratic equation. To solve for x, we can either factor, complete the square, or use the quadratic formula. In this case, factoring may not be the most straightforward method, so we'll use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 3, b = -17, and c = 14. Plugging these values into the quadratic formula:
x = (-(-17) ± √((-17)^2 - 4 * 3 * 14)) / (2 * 3)
Simplifying the formula:
x = (17 ± √(289 - 168)) / 6
x = (17 ± √(121)) / 6
x = (17 ± 11) / 6
This gives us two potential solutions:
x1 = (17 + 11) / 6 = 28/6 = 14/3
x2 = (17 - 11) / 6 = 6/6 = 1
Therefore, the solutions to the equation are x = 14/3 and x = 1.