GATE CS 2014 Set 1 Q.5 (GA Q.5): Quantitative Aptitude

The roots of $ax^2 + bx + c = 0$ are real and positive. $a$ , $b$ and $c$ are real. Then $ax^2 + b|x| + c = 0$ has

no roots

2 real roots

3 real roots

4 real roots

Official answer and solution

Official answer: D
From: Official GATE 2014 answer key

Solution

Relate the roots of $ax^2 + b|x| + c = 0$ to the roots of $ax^2 + bx + c = 0$ using the properties of absolute value.

We are given that the quadratic equation $ax^2 + bx + c = 0$ has real and positive roots. Let these distinct roots be $r_1$ and $r_2$ , where $r_1 > 0$ and $r_2 > 0$ .

Now consider the equation:

ax^2 + b|x| + c = 0

Since $x^2 = |x|^2$ for all real $x$ , this equation can be rewritten as:

a(|x|)^2 + b|x| + c = 0

This is a quadratic equation in $|x|$ . Therefore, the values of $|x|$ must be the roots of the original quadratic equation:

|x| = r_1 \quad \text{or} \quad |x| = r_2

Since both $r_1$ and $r_2$ are strictly positive real numbers:

$|x| = r_1$ yields two distinct real solutions: $x = r_1$ and $x = -r_1$ .
$|x| = r_2$ yields two distinct real solutions: $x = r_2$ and $x = -r_2$ .

Since $r_1 \neq r_2$ , all four values $\{r_1, -r_1, r_2, -r_2\}$ are distinct real numbers. Hence, the equation has 4 real roots.

Final answer

Common trap

Assuming $|x|$ can only yield positive values of $x$ , forgetting that $x$ itself can be negative.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2014 Computer Science and Information Technology (CS) Session 1
Question: 5 of 65, printed as GA Q.5
Section: General Aptitude

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