Question

Let a, b $\in$ C. Let $\alpha, \beta$ be the roots of the equation $x^2 + ax + b = 0$. If $\beta-\alpha =\sqrt{11}$ and $\beta^2-\alpha^2 = 3i\sqrt{11}$, then $(\beta^3 - \alpha^3)^2$ is equal to:

• A. 160
• B. 176
• C. 194
• D. 187

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given information about the difference and the difference of squares of the roots of a quadratic equation. Our objective is to use this information to calculate the value of $(\beta^3 - \alpha^3)^2$.
Step 2: Key Formula or Approach:
The strategy is to find the elementary symmetric polynomials, the sum of roots ($\alpha + \beta$) and the product of roots ($\alpha\beta$), using the given data. Then, we will express the target expression, $(\beta^3 - \alpha^3)^2$, in terms of these quantities.
The essential algebraic identities are:
1. $\beta^2 - \alpha^2 = (\beta - \alpha)(\beta + \alpha)$
2. $(\beta + \alpha)^2 - (\beta - \alpha)^2 = 4\alpha\beta$
3. $\beta^3 - \alpha^3 = (\beta - \alpha)((\beta + \alpha)^2 - \alpha\beta)$
Step 3: Detailed Explanation:
We are provided with:
\[ \beta - \alpha = \sqrt{11} \quad \text{---(i)} \] \[ \beta^2 - \alpha^2 = 3i\sqrt{11} \quad \text{---(ii)} \] First, let's find the sum of the roots using the identity for the difference of squares:
\[ \beta^2 - \alpha^2 = (\beta - \alpha)(\beta + \alpha) \] Substituting the given values:
\[ 3i\sqrt{11} = (\sqrt{11})(\beta + \alpha) \] Dividing both sides by $\sqrt{11}$ gives us the sum of the roots:
\[ \beta + \alpha = 3i \quad \text{---(iii)} \] Next, we find the product of the roots, $\alpha\beta$, using the identity that connects the sum and difference:
\[ (\beta + \alpha)^2 - (\beta - \alpha)^2 = 4\alpha\beta \] Substituting the known values from (i) and (iii):
\[ (3i)^2 - (\sqrt{11})^2 = 4\alpha\beta \] \[ 9i^2 - 11 = 4\alpha\beta \] As $i^2 = -1$:
\[ -9 - 11 = 4\alpha\beta \] \[ -20 = 4\alpha\beta \implies \alpha\beta = -5 \] Now, we can compute $\beta^3 - \alpha^3$ using its standard factorization:
\[ \beta^3 - \alpha^3 = (\beta - \alpha)((\beta + \alpha)^2 - \alpha\beta) \] Plugging in all the values we've found:
\[ \beta^3 - \alpha^3 = (\sqrt{11})((3i)^2 - (-5)) \] \[ \beta^3 - \alpha^3 = \sqrt{11}(-9 + 5) = \sqrt{11}(-4) = -4\sqrt{11} \] Finally, we square this result to get the answer:
\[ (\beta^3 - \alpha^3)^2 = (-4\sqrt{11})^2 = 16 \times 11 = 176 \] Step 4: Final Answer:
The value of $(\beta^3 - \alpha^3)^2$ is 176.