Question

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

• 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 1

Answer: (C)

Since, $\alpha$ and $\beta$ be the roots of the equation
$ax^2 + bx + c = 0$, then
$\alpha + \beta = -\frac{b}{a}$ and $\alpha\beta = \frac{c}{a} $
Now, sum of roots $= \left|\alpha\right|+\left|\beta\right|$
$= -\alpha - \beta\quad\left(\because\quad\beta < \alpha < 0\right)$
$ = -\left(-\frac{b}{a}\right) = \left|\frac{b}{a}\right| \quad\left(\because\quad\left|\alpha \right|+\left|\beta \right| > 0\right)$
and product of roots $= \left|\alpha \right|\left|\beta \right| = \left|\frac{c}{a}\right|$
Hence, required equation is
$x^{2}-\left|\frac{b}{a}\right|x+\left|\frac{c}{a}\right| = 0$
$\Rightarrow \quad|a| x^{2} - |b| x + |c| = 0$