Question

The number of distinct real roots of the equation \(\bigg(\frac{x+1}{x}\bigg)^2-3\bigg(\frac{x+1}{x}\bigg)+2=0\) equals

Answer 1

We have an equation containing the expression: \(\frac{x+1}{x}\). Let's perform a substitution:

Let: \(\frac{x+1}{x} = a\)

The original equation transforms into:

\[a^2 - 3a + 2 = 0\]

Solving this quadratic equation yields:

\[a = 1 \quad \text{or} \quad a = 2\]

Analysis of Each Value of \(a\)

• Case 1: \(a = 2 \Rightarrow \frac{x+1}{x} = 2\)
  Multiplying both sides by \(x\) gives: \(x + 1 = 2x \Rightarrow x = 1\)
  This is a valid real solution ✅
• Case 2: \(a = 1 \Rightarrow \frac{x+1}{x} = 1\)
  Multiplying both sides by \(x\) gives: \(x + 1 = x \Rightarrow 1 = 0\)
  ❌ This is a contradiction, hence not a valid solution.

 Conclusion

Therefore, the sole real solution is: \(x = 1\)
Total number of real solutions = 1