If ` alpha and beta ` are roots of the equation ` a x^(2) + b x + c = 0 ` then which equation will have roots `( alpha beta + alpha + beta ) and ( alpha beta - alpha - beta )` ?

Options

A. ` a^(2) x^(2) + 2acx + c^(2) + b^(2) = 0`
B. ` a^(2) x^(2) - 2 acx + c^(2) - b^(2) = 0 `
C. ` a^(2) x^(2) - 2acx + c^(2) + b^(2) = 0 `
D. ` a^(2) x^(2) + 2 acx + c^(2) - b^(2) = 0 `

Answer

To find the equation that has roots \( \alpha \beta + \alpha + \beta \) and \( \alpha \beta - \alpha - \beta \), we can follow these steps:

### Step 1: Identify the roots
Let:
- \( a = \alpha \beta + \alpha + \beta \)
- \( b = \alpha \beta - \alpha - \beta \)

### Step 2: Use Vieta's formulas
From the original quadratic 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 3: Calculate the new sum of roots
The sum of the new roots \( a + b \) is:
\[ a + b = (\alpha \beta + \alpha + \beta) + (\alpha \beta - \alpha - \beta) = 2\alpha \beta \]
Substituting \( \alpha \beta = \frac{c}{a} \):
\[ a + b = 2\left(\frac{c}{a}\right) = \frac{2c}{a} \]

### Step 4: Calculate the new product of roots
The product of the new roots \( a \cdot b \) is:
\[ a \cdot b = (\alpha \beta + \alpha + \beta)(\alpha \beta - \alpha - \beta) = (\alpha \beta)^2 - (\alpha + \beta)^2 \]
Using the identities:
\[ (\alpha + \beta)^2 = \left(-\frac{b}{a}\right)^2 = \frac{b^2}{a^2} \]
and
\[ (\alpha \beta)^2 = \left(\frac{c}{a}\right)^2 = \frac{c^2}{a^2} \]
Thus,
\[ a \cdot b = \frac{c^2}{a^2} - \frac{b^2}{a^2} = \frac{c^2 - b^2}{a^2} \]

### Step 5: Form the new quadratic equation
The new quadratic equation can be formed using the sum and product of the roots:
\[ x^2 - (a + b)x + (a \cdot b) = 0 \]
Substituting the values we found:
\[ x^2 - \frac{2c}{a}x + \frac{c^2 - b^2}{a^2} = 0 \]

### Step 6: Clear the denominators
Multiply through by \( a^2 \):
\[ a^2 x^2 - 2ac x + (c^2 - b^2) = 0 \]
Rearranging gives:
\[ a^2 x^2 - 2ac x + c^2 - b^2 = 0 \]

### Conclusion
The equation that has roots \( \alpha \beta + \alpha + \beta \) and \( \alpha \beta - \alpha - \beta \) is:
\[ a^2 x^2 - 2ac x + c^2 - b^2 = 0 \]
This corresponds to option B from the original question.