GATE CS 2014 Set 1 QGA-5 — General Aptitude

Let the roots of the quadratic equation $ax^2 + bx + c = 0$ be $\alpha$ and $\beta$.

We are given that $\alpha$ and $\beta$ are real and positive (i.e., $\alpha > 0$ and $\beta > 0$).

Now, consider the given equation: $$ax^2 + b|x| + c = 0$$

Since $x^2 = |x|^2$ for any real number $x$, we can rewrite the equation as: $$a|x|^2 + b|x| + c = 0$$

Let $y = |x|$. The equation becomes: $$ay^2 + by + c = 0$$

The roots of this quadratic equation in $y$ are the same as the roots of the original equation, which are $\alpha$ and $\beta$. Therefore: $$|x| = \alpha \quad \text{or} \quad |x| = \beta$$

Since both $\alpha$ and $\beta$ are positive real numbers, we solve for $x$:

• From $|x| = \alpha$, we get $x = \pm \alpha$ (2 real roots)
• From $|x| = \beta$, we get $x = \pm \beta$ (2 real roots)

Thus, the equation has a total of $4$ real roots (namely $\alpha, -\alpha, \beta, -\beta$).