Find the quadratic equation whose roots are the reciprocals of the roots of the equation `x^(2) - cx + b = 0 `

Answer

To find the quadratic equation whose roots are the reciprocals of the roots of the equation \( x^2 - cx + b = 0 \), we can follow these steps:

### Step 1: Identify the roots of the original equation
Let the roots of the equation \( x^2 - cx + b = 0 \) be \( \alpha \) and \( \beta \).

### Step 2: Use Vieta's formulas
According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = c \)
- The product of the roots \( \alpha \beta = b \)

### Step 3: Find the roots of the new equation
The roots we need to find are the reciprocals of the original roots, which are \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \).

### Step 4: Calculate the sum of the new roots
The sum of the new roots is:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta} = \frac{c}{b} \]

### Step 5: Calculate the product of the new roots
The product of the new roots is:
\[ \frac{1}{\alpha} \cdot \frac{1}{\beta} = \frac{1}{\alpha \beta} = \frac{1}{b} \]

### Step 6: Form the new quadratic equation
Using the standard form of a quadratic equation with roots \( r_1 \) and \( r_2 \):
\[ x^2 - (r_1 + r_2)x + r_1 r_2 = 0 \]
we substitute \( r_1 = \frac{1}{\alpha} \) and \( r_2 = \frac{1}{\beta} \):
\[ x^2 - \left(\frac{c}{b}\right)x + \frac{1}{b} = 0 \]

### Step 7: Clear the fractions
To eliminate the fractions, multiply the entire equation by \( b \):
\[ b x^2 - c x + 1 = 0 \]

### Final Result
Thus, the quadratic equation whose roots are the reciprocals of the roots of the equation \( x^2 - cx + b = 0 \) is:
\[ b x^2 - c x + 1 = 0 \]