Let `alpha,beta` be the roots of the equation `ax^2+bx+c=0`. A root of the equation `a^3x^2+abcx+c^3=0` is (i) `alpha+beta` (ii) `alpha^2+beta` (iii)`alpha^2-beta` (iv) `alpha^2beta`

Options

A. `alpha + beta`
B. `alpha^(2) +beta`
C. `alpha ^(2) -beta`
D. `alpha ^(2) beta`

Answer

To find a root of the equation \( a^3x^2 + abcx + c^3 = 0 \) given that \( \alpha \) and \( \beta \) are the roots of the equation \( ax^2 + bx + c = 0 \), we can follow these steps:

### Step 1: Identify the roots and their relationships
From the equation \( ax^2 + bx + c = 0 \), we know the following:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

### Step 2: Write the second equation
The second equation we are analyzing is \( a^3x^2 + abcx + c^3 = 0 \). We will denote its roots as \( \gamma \) and \( \theta \).

### Step 3: Find the sum and product of the roots of the second equation
For the equation \( a^3x^2 + abcx + c^3 = 0 \):
- The sum of the roots \( \gamma + \theta = -\frac{abc}{a^3} = -\frac{bc}{a^2} \)
- The product of the roots \( \gamma \theta = \frac{c^3}{a^3} = \left(\frac{c}{a}\right)^3 = \left(\alpha \beta\right)^3 \)

### Step 4: Substitute the known values
Substituting the values of \( \alpha + \beta \) and \( \alpha \beta \) into the expressions for \( \gamma + \theta \) and \( \gamma \theta \):
- \( \gamma + \theta = -\frac{bc}{a^2} = \left(-\frac{b}{a}\right)\left(\frac{c}{a}\right) = (\alpha + \beta)(\alpha \beta) \)
- \( \gamma \theta = \left(\alpha \beta\right)^3 \)

### Step 5: Find \( \gamma - \theta \)
Using the identity \( \gamma - \theta = \sqrt{(\gamma + \theta)^2 - 4\gamma \theta} \):
- Substitute the values we found:
\[ \gamma - \theta = \sqrt{\left(-\frac{bc}{a^2}\right)^2 - 4\left(\frac{c^3}{a^3}\right)} \]
This can be simplified further, but we are primarily interested in finding a specific root.

### Step 6: Identify the roots
From the previous calculations, we can deduce:
1. One root is \( \gamma = \alpha^2 \beta \)
2. The other root is \( \theta = \alpha \beta^2 \)

### Conclusion
Among the options provided:
(i) \( \alpha + \beta \)
(ii) \( \alpha^2 + \beta \)
(iii) \( \alpha^2 - \beta \)
(iv) \( \alpha^2 \beta \)

The correct answer is **(iv) \( \alpha^2 \beta \)**.

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