If `alpha and beta` are the roots of the equations `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)`

Answer

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: Identify the roots of the first equation
The roots \( \alpha \) and \( \beta \) of the equation \( ax^2 + bx + c = 0 \) can be expressed using Vieta's formulas:
- 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
We need to analyze the second equation \( cx^2 + bx + a = 0 \).

### Step 3: Use Vieta's formulas for the second equation
For the equation \( cx^2 + bx + a = 0 \), we can again apply Vieta's formulas:
- The sum of the roots \( r_1 + r_2 = -\frac{b}{c} \)
- The product of the roots \( r_1 r_2 = \frac{a}{c} \)

### Step 4: Relate the roots of the second equation to the roots of the first
To find the roots of \( cx^2 + bx + a = 0 \) in terms of \( \alpha \) and \( \beta \):
1. We know that \( \alpha + \beta = -\frac{b}{a} \).
2. We also know that \( \alpha \beta = \frac{c}{a} \).

Now, we can express the roots \( r_1 \) and \( r_2 \) of the second equation in terms of \( \alpha \) and \( \beta \):
- The sum of the roots \( r_1 + r_2 = -\frac{b}{c} \) can be rewritten using \( \alpha \) and \( \beta \).
- The product of the roots \( r_1 r_2 = \frac{a}{c} \) can also be rewritten using \( \alpha \) and \( \beta \).

### Step 5: Conclude the relationship
By comparing the sums and products from both equations, we can conclude that the roots \( r_1 \) and \( r_2 \) of the equation \( cx^2 + bx + a = 0 \) are indeed \( \alpha \) and \( \beta \).

### Final Answer
Thus, the roots of the equation \( cx^2 + bx + a = 0 \) are \( \alpha \) and \( \beta \).

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