If ($x + \frac{1}{x}$) = 2, then the value of ($\sqrt{x} + \frac{1}{\sqrt{x}}$) is :

Q: If ($x + \frac{1}{x}$) = 2, then the value of ($\sqrt{x} + \frac{1}{\sqrt{x}}$) is :

To solve the problem, we need to find the value of $(\sqrt{x} + \frac{1}{\sqrt{x}})$ given that $(x + \frac{1}{x}) = 2$. **Explanation:** 1. **Given Equation:** \[ x + \frac{1}{x} = 2 \] 2. **Objective:** We need to find $\sqrt{x} + \frac{1}{\sqrt{x}}$. 3. **Analysis:** Let's first consider the given equation $x + \frac{1}{x} = 2$. This equation implies that $x$ is a number such that when added to its reciprocal, the result is 2. 4. **Solving for $x$:** - Multiply both sides by $x$ to eliminate the fraction: \[ x^2 + 1 = 2x \] - Rearrange the equation: \[ x^2 - 2x + 1 = 0 \] - This can be factored as: \[ (x - 1)^2 = 0 \] - Solving this gives: \[ x = 1 \] 5. **Finding $\sqrt{x} + \frac{1}{\sqrt{x}}$:** - Since $x = 1$, we have: \[ \sqrt{x} = \sqrt{1} = 1 \] - Therefore: \[ \frac{1}{\sqrt{x}} = \frac{1}{1} = 1 \] - Thus: \[ \sqrt{x} + \frac{1}{\sqrt{x}} = 1 + 1 = 2 \] 6. **Conclusion:** The value of $\sqrt{x} + \frac{1}{\sqrt{x}}$ is indeed 2, which corresponds to Option 2. **Why Other Options are Incorrect:** - **Option 1: $\sqrt{2}$** - This option is incorrect because the calculations show that $\sqrt{x} + \frac{1}{\sqrt{x}} = 2$, not $\sqrt{2}$. - **Option 3: 1** - This option is incorrect because both $\sqrt{x}$ and $\frac{1}{\sqrt{x}}$ are 1, and their sum is 2, not 1. - **Option 4: $1 + \sqrt{2}$** - This option is incorrect because the sum of $\sqrt{x}$ and $\frac{1}{\sqrt{x}}$ is exactly 2, not $1 + \sqrt{2}$. Therefore, the correct answer is Option 2, as the calculations and logical reasoning confirm that $\sqrt{x} + \frac{1}{\sqrt{x}} = 2$.

Question

Accepted Answer

To solve the problem, we need to find the value of $(\sqrt{x} + \frac{1}{\sqrt{x}})$ given that $(x + \frac{1}{x}) = 2$.

**Explanation:**

1. **Given Equation:**
   $$
   x + \frac{1}{x} = 2
   $$

2. **Objective:**
   We need to find $\sqrt{x} + \frac{1}{\sqrt{x}}$.

3. **Analysis:**
   Let's first consider the given equation $x + \frac{1}{x} = 2$. This equation implies that $x$ is a number such that when added to its reciprocal, the result is 2.

4. **Solving for $x$:**
   - Multiply both sides by $x$ to eliminate the fraction:
     $$
     x^2 + 1 = 2x
     $$
   - Rearrange the equation:
     $$
     x^2 - 2x + 1 = 0
     $$
   - This can be factored as:
     $$
     (x - 1)^2 = 0
     $$
   - Solving this gives:
     $$
     x = 1
     $$

5. **Finding $\sqrt{x} + \frac{1}{\sqrt{x}}$:**
   - Since $x = 1$, we have:
     $$
     \sqrt{x} = \sqrt{1} = 1
     $$
   - Therefore:
     $$
     \frac{1}{\sqrt{x}} = \frac{1}{1} = 1
     $$
   - Thus:
     $$
     \sqrt{x} + \frac{1}{\sqrt{x}} = 1 + 1 = 2
     $$

6. **Conclusion:**
   The value of $\sqrt{x} + \frac{1}{\sqrt{x}}$ is indeed 2, which corresponds to Option 2.

**Why Other Options are Incorrect:**

- **Option 1: $\sqrt{2}$**
  - This option is incorrect because the calculations show that $\sqrt{x} + \frac{1}{\sqrt{x}} = 2$, not $\sqrt{2}$.

- **Option 3: 1**
  - This option is incorrect because both $\sqrt{x}$ and $\frac{1}{\sqrt{x}}$ are 1, and their sum is 2, not 1.

- **Option 4: $1 + \sqrt{2}$**
  - This option is incorrect because the sum of $\sqrt{x}$ and $\frac{1}{\sqrt{x}}$ is exactly 2, not $1 + \sqrt{2}$.

Therefore, the correct answer is Option 2, as the calculations and logical reasoning confirm that $\sqrt{x} + \frac{1}{\sqrt{x}} = 2$.