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

Answer

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

### Step 1: Identify the roots and their properties
Given that \( \alpha \) and \( \beta \) are the roots of the equation \( ax^2 + bx + c = 0 \), we know:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

### Step 2: Find the sum of the new roots
The new roots are \( \alpha + 1 \) and \( \beta + 1 \). Therefore, we can calculate their sum:
\[ (\alpha + 1) + (\beta + 1) = \alpha + \beta + 2 \]
Substituting the value of \( \alpha + \beta \):
\[ \alpha + \beta + 2 = -\frac{b}{a} + 2 = -\frac{b}{a} + \frac{2a}{a} = \frac{-b + 2a}{a} \]

### Step 3: Find the product of the new roots
Next, we calculate the product of the new roots:
\[ (\alpha + 1)(\beta + 1) = \alpha \beta + \alpha + \beta + 1 \]
Substituting the values of \( \alpha + \beta \) and \( \alpha \beta \):
\[ \alpha \beta + \alpha + \beta + 1 = \frac{c}{a} + \left(-\frac{b}{a}\right) + 1 = \frac{c - b + a}{a} \]

### Step 4: Form the new quadratic equation
Using the sum and product of the new roots, we can form the quadratic equation:
\[ x^2 - \text{(sum of roots)} \cdot x + \text{(product of roots)} = 0 \]
Substituting the values we found:
\[ x^2 - \left(\frac{-b + 2a}{a}\right)x + \left(\frac{c - b + a}{a}\right) = 0 \]

### Step 5: Clear the denominators
To eliminate the fraction, multiply the entire equation by \( a \):
\[ a x^2 - (-b + 2a)x + (c - b + a) = 0 \]
This simplifies to:
\[ a x^2 + (b - 2a)x + (c - b + a) = 0 \]

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

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