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.