If α and ẞ are the roots of the equation $x^2 -2x -1=0$, then the value of (α$^2$β + αβ$^2$) is :

Question

Accepted Answer

To solve the problem, we need to find the value of $ \alpha^2 \beta + \alpha \beta^2 $ given that $ \alpha $ and $ \beta $ are the roots of the quadratic equation $ x^2 - 2x - 1 = 0 $.

### Step-by-step Explanation:

1. **Identify the Roots:**
   The roots $ \alpha $ and $ \beta $ of the quadratic equation $ x^2 - 2x - 1 = 0 $ can be found using the quadratic formula:
   $$
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   $$
   Here, $ a = 1 $, $ b = -2 $, and $ c = -1 $. Plugging these values into the formula gives:
   $$
   x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{2 \pm \sqrt{4 + 4}}{2} = \frac{2 \pm \sqrt{8}}{2} = \frac{2 \pm 2\sqrt{2}}{2}
   $$
   Simplifying, we get the roots:
   $$
   \alpha = 1 + \sqrt{2}, \quad \beta = 1 - \sqrt{2}
   $$

2. **Calculate $ \alpha \beta $:**
   Using Vieta's formulas, the product of the roots $ \alpha \beta $ is equal to the constant term divided by the leading coefficient:
   $$
   \alpha \beta = c/a = -1/1 = -1
   $$

3. **Calculate $ \alpha^2 \beta + \alpha \beta^2 $:**
   We can express $ \alpha^2 \beta + \alpha \beta^2 $ in terms of $ \alpha \beta $ and $ \alpha + \beta $:
   $$
   \alpha^2 \beta + \alpha \beta^2 = \alpha \beta (\alpha + \beta)
   $$
   From Vieta's formulas, the sum of the roots $ \alpha + \beta = 2 $.

4. **Substitute the Known Values:**
   Substitute $ \alpha \beta = -1 $ and $ \alpha + \beta = 2 $ into the expression:
   $$
   \alpha^2 \beta + \alpha \beta^2 = (-1)(2) = -2
   $$

Thus, the value of $ \alpha^2 \beta + \alpha \beta^2 $ is $-2$, which corresponds to **Option 1**.

### Explanation of Incorrect Options:

- **Option 2 (3):** This option is incorrect because the calculation of $ \alpha^2 \beta + \alpha \beta^2 $ using the correct values of $ \alpha \beta $ and $ \alpha + \beta $ results in $-2$, not 3.

- **Option 3 (2):** This option is incorrect for the same reason as above. The correct computation yields $-2$, not 2.

- **Option 4 (1):** This option is incorrect because the correct value of the expression is $-2$, not 1.

In conclusion, the correct answer is indeed **Option 1 (-2)**, as derived from the calculations based on the properties of the roots of the quadratic equation.