If `alpha,beta` are the roots of the equation `ax^2 + bx +c=0` such that `beta < alpha < 0`, then the quadratic equation whose roots are `|alpha|,|beta|` is given by

Options

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

Answer

To find the quadratic equation whose roots are \(|\alpha|\) and \(|\beta|\), given that \(\alpha\) and \(\beta\) are the roots of the equation \(ax^2 + bx + c = 0\) with the conditions \(\beta < \alpha < 0\), we can follow these steps:

### Step-by-Step Solution:

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

2. **Analyze the signs of the roots**:
- Since \(\beta < \alpha < 0\), both roots are negative. Therefore:
- \(\alpha + \beta < 0\) implies \(-\frac{b}{a} < 0\), which means \(b/a > 0\). This indicates that \(a\) and \(b\) must have the same sign (both positive or both negative).
- \(\alpha \beta > 0\) implies \(\frac{c}{a} > 0\), which means \(c/a > 0\). This indicates that \(c\) must also have the same sign as \(a\).

3. **Calculate the new roots**:
- The new roots we are interested in are \(|\alpha|\) and \(|\beta|\). Since both \(\alpha\) and \(\beta\) are negative:
- \(|\alpha| = -\alpha\)
- \(|\beta| = -\beta\)

4. **Find the sum and product of the new roots**:
- The sum of the new roots:
\[ |\alpha| + |\beta| = -\alpha - \beta = -(\alpha + \beta) = \frac{b}{a} \]
- The product of the new roots:
\[ |\alpha| \cdot |\beta| = (-\alpha)(-\beta) = \alpha \beta = \frac{c}{a} \]

5. **Form the quadratic equation**:
- The quadratic equation with roots \(|\alpha|\) and \(|\beta|\) can be expressed as:
\[ x^2 - (|\alpha| + |\beta|)x + |\alpha| \cdot |\beta| = 0 \]
- Substituting the values of the sum and product:
\[ x^2 - \left(\frac{b}{a}\right)x + \left(\frac{c}{a}\right) = 0 \]

6. **Multiply through by \(a\)** to eliminate the fraction:
\[ ax^2 - bx + c = 0 \]

7. **Express in terms of absolute values**:
- Since we need the equation in terms of absolute values:
\[ \text{The required quadratic equation is } |a|x^2 - |b|x + |c| = 0 \]

### Final Answer:
The quadratic equation whose roots are \(|\alpha|\) and \(|\beta|\) is:
\[ |a|x^2 - |b|x + |c| = 0 \]