Solving the Equation to Find the Value of an Expression

We are given an equation involving the expression \( \left( \frac{x}{x+1} \right) \) and we need to find the value of \( \left( 1+\frac{1}{x} \right) \). The given equation is:

\( \left( \frac{x}{x+1} \right)^2 -5 \left( \frac{x}{x+1} \right) +6=0 \)

Simplifying the Equation with Substitution

This equation looks like a quadratic equation. We can make it simpler by using a substitution. Let \( y = \frac{x}{x+1} \). Substituting \( y \) into the equation gives us a standard quadratic form:

\( y^2 - 5y + 6 = 0 \)

Solving the Quadratic Equation for y

We can solve this quadratic equation by factoring. We need two numbers that multiply to 6 and add up to -5. These numbers are -2 and -3.

So, the equation can be factored as:

\( (y - 2)(y - 3) = 0 \)

This gives us two possible values for \( y \):

• \( y - 2 = 0 \implies y = 2 \)
• \( y - 3 = 0 \implies y = 3 \)

Finding the Value of the Required Expression

Now we need to find the value of \( \left( 1+\frac{1}{x} \right) \). Let's look at this expression and see how it relates to \( \frac{x}{x+1} \).

\( 1+\frac{1}{x} = \frac{x}{x} + \frac{1}{x} = \frac{x+1}{x} \)

Notice that \( \frac{x+1}{x} \) is the reciprocal of \( \frac{x}{x+1} \). So, if \( y = \frac{x}{x+1} \), then \( \left( 1+\frac{1}{x} \right) = \frac{1}{y} \).

Now we can find the possible values for \( \left( 1+\frac{1}{x} \right) \) using the values we found for \( y \):

• If \( y = 2 \), then \( \left( 1+\frac{1}{x} \right) = \frac{1}{y} = \frac{1}{2} \).
• If \( y = 3 \), then \( \left( 1+\frac{1}{x} \right) = \frac{1}{y} = \frac{1}{3} \).

Thus, the possible values for \( \left( 1+\frac{1}{x} \right) \) are \( \frac{1}{2} \) or \( \frac{1}{3} \).