The value of x which satisfy the equation $5^{1+x}+5^{1-x}=26$ are :

Question

Accepted Answer

To solve the equation $5^{1+x} + 5^{1-x} = 26$, we need to find the values of $x$ that satisfy this equation. Let's go through the solution step by step to justify why Option 1 (-1, 1) is correct.

### Step-by-Step Solution:

1. **Substitution and Simplification:**
   - Let $y = 5^x$. Then $5^{1+x} = 5 \cdot 5^x = 5y$ and $5^{1-x} = \frac{5}{5^x} = \frac{5}{y}$.
   - Substitute these into the equation: 
     $$
     5y + \frac{5}{y} = 26
     $$

2. **Clear the Fraction:**
   - Multiply the entire equation by $y$ to eliminate the fraction:
     $$
     5y^2 + 5 = 26y
     $$

3. **Rearrange the Equation:**
   - Rearrange the terms to form a quadratic equation:
     $$
     5y^2 - 26y + 5 = 0
     $$

4. **Solve the Quadratic Equation:**
   - Use the quadratic formula $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 5$, $b = -26$, and $c = 5$:
     $$
     y = \frac{26 \pm \sqrt{(-26)^2 - 4 \cdot 5 \cdot 5}}{2 \cdot 5}
     $$
     $$
     y = \frac{26 \pm \sqrt{676 - 100}}{10}
     $$
     $$
     y = \frac{26 \pm \sqrt{576}}{10}
     $$
     $$
     y = \frac{26 \pm 24}{10}
     $$

5. **Calculate the Roots:**
   - The roots are:
     $$
     y = \frac{26 + 24}{10} = \frac{50}{10} = 5
     $$
     $$
     y = \frac{26 - 24}{10} = \frac{2}{10} = 0.2
     $$

6. **Find $x$ from $y$:**
   - Recall $y = 5^x$. Therefore:
     - For $y = 5$, $5^x = 5$ implies $x = 1$.
     - For $y = 0.2$, $5^x = 0.2$ implies $x = -1$ (since $5^{-1} = 0.2$).

### Conclusion:

The values of $x$ that satisfy the equation are $x = -1$ and $x = 1$, which corresponds to Option 1 (-1, 1).

### Explanation of Incorrect Options:

- **Option 2 (0, 1):** 
  - $x = 0$ would imply $5^{1+0} + 5^{1-0} = 5 + 5 = 10$, which does not satisfy the equation.
  
- **Option 3 (1, 2):**
  - $x = 2$ would imply $5^{1+2} + 5^{1-2} = 125 + 0.2 = 125.2$, which does not satisfy the equation.
  
- **Option 4 (-1, 0):**
  - As shown above, $x = 0$ does not satisfy the equation.

Thus, the correct answer is indeed Option 1 (-1, 1), as it is the only option that satisfies the given equation.