If`alpha and beta` are the roots of the equation `ax^(2)+bx+c=0,` then what are the roots of the equation `cx^(2) + bx + a = 0 ` ?

Options

A. ` beta , 1/alpha`
B. ` alpha, 1/beta`
C. ` -alpha, - beta`
D. ` 1/alpha, 1/beta`

Explanation

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

### Step 1: Understand the given information
We know that \( \alpha \) and \( \beta \) are the roots of the equation \( ax^2 + bx + c = 0 \). From Vieta's formulas, we can derive the following:

- Sum of the roots:
\[ \alpha + \beta = -\frac{b}{a} \]

- Product of the roots:
\[ \alpha \beta = \frac{c}{a} \]

### Step 2: Analyze the new equation
Now, we need to analyze the equation \( cx^2 + bx + a = 0 \). Again, using Vieta's formulas, we can find the sum and product of the roots of this equation:

- Sum of the roots (let's call them \( p \) and \( q \)):
\[ p + q = -\frac{b}{c} \]

- Product of the roots:
\[ pq = \frac{a}{c} \]

### Step 3: Relate the roots of the new equation to the original roots
We will express the sum and product of the roots \( p \) and \( q \) in terms of \( \alpha \) and \( \beta \).

1. **Sum of the roots**:
We have:
\[ p + q = -\frac{b}{c} \]
From the first equation, we know that:
\[ \alpha + \beta = -\frac{b}{a} \]
To relate these, we can multiply \( \alpha + \beta \) by \( \frac{a}{c} \):
\[ \frac{a}{c}(\alpha + \beta) = -\frac{b}{c} \]

Thus, we can conclude:
\[ p + q = \frac{a}{c}(\alpha + \beta) \]

2. **Product of the roots**:
We have:
\[ pq = \frac{a}{c} \]
From the product of the original roots, we know:
\[ \alpha \beta = \frac{c}{a} \]
To relate these, we can multiply \( \alpha \beta \) by \( \frac{c}{a} \):
\[ \frac{c}{a}(\alpha \beta) = \frac{a}{c} \]

### Step 4: Conclusion
From the relationships we derived, we can conclude that the roots \( p \) and \( q \) of the equation \( cx^2 + bx + a = 0 \) are related to the roots \( \alpha \) and \( \beta \) of the original equation \( ax^2 + bx + c = 0 \) as follows:

If \( \alpha \) and \( \beta \) are the roots of the first equation, then the roots of the second equation \( cx^2 + bx + a = 0 \) are \( \frac{c}{\alpha} \) and \( \frac{c}{\beta} \).