If `alpha " & " beta` are the roots of equation `ax^(2) + bx + c = 0` then find the Quadratic equation whose roots are `(1)/(alpha) " & " (1)/(beta)`

Answer

To find the quadratic equation whose roots are \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \), we can follow these steps:

### Step 1: Identify the sum and product of the original roots
The roots of the original quadratic equation \( ax^2 + bx + c = 0 \) are \( \alpha \) and \( \beta \).
From Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

### Step 2: Calculate the sum of the new roots
The new roots are \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \).
The sum of these new roots can be calculated as follows:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta} \]
Substituting the values from Step 1:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{-\frac{b}{a}}{\frac{c}{a}} = -\frac{b}{c} \]

### Step 3: Calculate the product of the new roots
The product of the new roots \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \) is given by:
\[ \frac{1}{\alpha} \cdot \frac{1}{\beta} = \frac{1}{\alpha \beta} \]
Substituting the value from Step 1:
\[ \frac{1}{\alpha} \cdot \frac{1}{\beta} = \frac{1}{\frac{c}{a}} = \frac{a}{c} \]

### Step 4: Form the new quadratic equation
Using the sum and product of the new roots, we can now form the quadratic equation:
\[ x^2 - \left(\text{sum of roots}\right)x + \left(\text{product of roots}\right) = 0 \]
Substituting the values we calculated:
\[ x^2 - \left(-\frac{b}{c}\right)x + \frac{a}{c} = 0 \]
This simplifies to:
\[ x^2 + \frac{b}{c}x + \frac{a}{c} = 0 \]
To eliminate the fraction, we can multiply through by \( c \):
\[ cx^2 + bx + a = 0 \]

### Final Result
The quadratic equation whose roots are \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \) is:
\[ cx^2 + bx + a = 0 \]

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