If `alpha , beta ` are the roots of the quadratic equation `ax^(2) + bx + c = 0` , then the quadratic equation whose roots are `alpha^(3) , beta^(3)` is

Options

A. `a^(3) y^(3) + (b^(3) - 3abc) y + c^(3) = 0`
B. `a^(3) y^(2) + (3abc - b^(3)) y - c^(3) = 0`
C. `a^(2) y^(2) + 2aby + c^(2) = 0`
D. none of these

Answer

To find the quadratic equation whose roots are \( \alpha^3 \) and \( \beta^3 \), we will follow these steps:

### Step 1: Find the sum and product of the roots \( \alpha \) and \( \beta \)
For the quadratic equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( \alpha + \beta \) is given by:
\[ \alpha + \beta = -\frac{b}{a} \]
- The product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{c}{a} \]

### Step 2: Calculate \( \alpha^3 + \beta^3 \)
Using the identity:
\[ \alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 - \alpha \beta + \beta^2) \]
We can express \( \alpha^2 + \beta^2 \) in terms of \( \alpha + \beta \) and \( \alpha \beta \):
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Thus,
\[ \alpha^3 + \beta^3 = (\alpha + \beta)((\alpha + \beta)^2 - 3\alpha\beta) \]
Substituting the values from Step 1:
\[ \alpha^3 + \beta^3 = \left(-\frac{b}{a}\right) \left(\left(-\frac{b}{a}\right)^2 - 3\frac{c}{a}\right) \]

### Step 3: Calculate \( \alpha^3 \beta^3 \)
The product \( \alpha^3 \beta^3 \) can be expressed as:
\[ \alpha^3 \beta^3 = (\alpha \beta)^3 = \left(\frac{c}{a}\right)^3 \]

### Step 4: Form the new quadratic equation
The new quadratic equation with roots \( \alpha^3 \) and \( \beta^3 \) can be written in the standard form:
\[ x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \]
Substituting the values we found:
\[ x^2 - \left(-\frac{b}{a}\left(\left(-\frac{b}{a}\right)^2 - 3\frac{c}{a}\right)\right)x + \left(\frac{c}{a}\right)^3 = 0 \]
This simplifies to:
\[ x^2 + \left(\frac{b^2 - 3ac}{a^2}\right)x + \frac{c^3}{a^3} = 0 \]

### Final Form of the Quadratic Equation
Thus, the quadratic equation whose roots are \( \alpha^3 \) and \( \beta^3 \) is:
\[ x^2 + \frac{b^2 - 3ac}{a^2} x + \frac{c^3}{a^3} = 0 \]