(2x-1)/(x+1)=2X/(X-1)+ (5/X)
To solve the equation (2x-1)/(x+1) = 2x/(x-1) + 5/x, we will use the following steps:
Step 1: Clear the denominators
To eliminate the denominators, we can multiply both sides of the equation by (x+1)(x-1)x. This will result in each term having a common denominator.
(x+1)(x-1)x * [(2x-1)/(x+1)] = (x+1)(x-1)x * [2x/(x-1) + 5/x]
Step 2: Expand and simplify
Expanding and simplifying both sides of the equation will help us combine like terms and reduce the equation to a more manageable form.
(x+1)(x-1)x * [(2x-1)/(x+1)] = (x+1)(x-1)x * [2x/(x-1) + 5/x]
(x-1)x(2x-1) = (x+1)(x-1)x * 2x + (x+1)(x-1)x * 5/x
Step 3: Distribute and simplify further
By distributing the terms on the right side of the equation, we can simplify each side of the equation.
(x-1)x(2x-1) = 2x(x+1)(x-1) + 5(x+1)(x-1)
2x^3 - 3x^2 + x = 2x(x^2 - 1) + 5(x^2 - 1)
Step 4: Simplify and combine like terms
Using the distributive property and combining like terms will further simplify the equation.
2x^3 - 3x^2 + x = 2x^3 - 2x + 5x^2 - 5
2x^3 - 3x^2 + x = 2x^3 + 5x^2 - 2x - 5
Step 5: Rearrange the terms
By rearranging the terms on both sides of the equation, we can bring similar terms together.
2x^3 - 3x^2 + x - 2x^3 - 5x^2 + 2x + 5 = 0
-8x^2 - 2x + 5 = 0
Step 6: Solve the quadratic equation
Now that we have a quadratic equation, we can solve it by factoring, completing the square, or using the quadratic formula.
Unfortunately, the equation -8x^2 - 2x + 5 = 0 does not factor easily. Completing the square can be time-consuming, so we'll use the quadratic formula to find the values of x.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our quadratic equation -8x^2 - 2x + 5 = 0:
a = -8, b = -2, c = 5
Plugging in these values into the quadratic formula, we get:
x = (-(-2) ± √((-2)^2 - 4(-8)(5))) / (2(-8))
Simplifying further:
x = (2 ± √(4 + 160)) / (-16)
x = (2 ± √(164)) / (-16)
x = (2 ± 2√(41)) / (-16)
x = (1 ± √(41)) / (-8)
So, the solution to the equation (2x-1)/(x+1) = 2x/(x-1) + 5/x is x = (1 ± √(41)) / (-8).