Cannot get this solution!!! (x-2)/(x) + (1/2)= (x+1)/(x-2)
(x-2)/(x) + (1/2)= (x+1)/(x-2)
Assume x≠0 and x≠2.
Add the two terms on the left using the common denominator of x.
[2(x-2)+x]/2x = (x+1)/(x-2)
(3x-4)/2x = (x+1)/(x-2)
Cross multiply:
(3x-4)(x-2) = 2x(x+1)
3x²-10x+8 = x²+x
x²-12x+8 = 0
x=(12±sqrt(112))/2
Simplifies to
x=6±2sqrt(7)
To solve the equation (x-2)/(x) + (1/2) = (x+1)/(x-2), we need to find the value of x that satisfies this equation.
Let's start by simplifying the equation.
First, we can simplify the left side by finding a common denominator for the fractions (x-2)/(x) and (1/2). The common denominator is 2x.
(x-2)/(x) + (1/2)
= (2(x-2))/(2x) + (x/2x)
= (2x - 4)/(2x) + (x)/(2x)
Now, let's simplify the right side of the equation by finding the common denominator for the fractions (x+1)/(x-2).
(x+1)/(x-2)
= ((x+1))/(x-2)
Now that we have common denominators on both sides of the equation, we can combine the fractions.
(2x - 4)/(2x) + (x)/(2x) = ((x+1))/(x-2)
Combining the fractions on the left side gives us:
(2x - 4 + x)/(2x) = ((x+1))/(x-2)
Simplifying the equation further:
(3x - 4)/(2x) = ((x+1))/(x-2)
To get rid of the denominators, we can cross-multiply.
Cross-multiplying gives us:
(3x - 4)(x-2) = 2x(x+1)
Expanding both sides of the equation:
3x^2 - 10x + 8 = 2x^2 + 2x
Now, we want to simplify this equation further by moving all the terms to one side and setting it equal to zero.
Subtracting 2x^2 and 2x from both sides:
3x^2 - 10x + 8 - 2x^2 - 2x = 0
Combining like terms:
x^2 - 12x + 8 = 0
Now, we have a quadratic equation. We can solve this equation by factoring, completing the square, or using the quadratic formula. However, upon inspection, it seems that it cannot be easily factored. Therefore, we'll use the quadratic formula to find the value(s) of x that satisfy this equation.
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac))/(2a)
For our equation x^2 - 12x + 8 = 0, a = 1, b = -12, and c = 8.
Using these values in the quadratic formula:
x = (-(-12) ± √((-12)^2 - 4(1)(8)))/(2(1))
= (12 ± √(144 - 32))/(2)
= (12 ± √(112))/2
= (12 ± 4√7)/2
= 6 ± 2√7
So, the two possible solutions for x are x = 6 + 2√7 and x = 6 - 2√7.