Solve 1/x +1/x+1=1 ?
assuming you meant
1/x + 1/(x+1) = 1
x+1 + x = x(x+1)
x^2-x-1 = 0
...
Thanks
To solve the equation 1/x + 1/(x+1) = 1, we need to find the value(s) of x that satisfy this equation.
To begin, we'll take the least common denominator (LCD) of the fractions, which is x(x+1). Multiplying each term on both sides of the equation by the LCD will eliminate the denominators.
Let's go through the steps:
1. Multiply each term on both sides of the equation by x(x+1):
[x(x+1)] * [1/x] + [x(x+1)] * [1/(x+1)] = [x(x+1)] * 1
Simplifying the equation gives:
(x+1) + x = x(x+1)
2. Expand the equation:
x + 1 + x = x^2 + x
Combine like terms:
2x + 1 = x^2 + x
3. Rearrange the equation to bring all terms to one side:
x^2 + x - 2x - 1 = 0
Simplify:
x^2 - x - 1 = 0
4. Now, we have a quadratic equation. We can solve it using various methods, such as factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
For our equation, a = 1, b = -1, and c = -1.
Plugging in these values:
x = (1 ± sqrt((-1)^2 - 4(1)(-1))) / (2(1))
Simplifying:
x = (1 ± sqrt(1 + 4)) / 2
x = (1 ± sqrt(5)) / 2
Therefore, the solutions to the equation are:
x = (1 + sqrt(5)) / 2 or x = (1 - sqrt(5)) / 2
These are the two possible values for x that satisfy the equation 1/x + 1/(x+1) = 1.