Problem 2 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(a, b, c\) are positive real numbers, then the number of real roots of the equation \(a x^{2}+b|x|+c=0\) is (A) 0 (B) 2 (C) 4 (D) None of these

Step-by-Step Solution

Answer

The equation can have 4 real roots.

Step 1: Understand the Equation

The given equation is \(a x^2 + b |x| + c = 0\), where \(a, b, c\) are positive real numbers. We need to find the number of real roots of this equation.

Step 2: Split the Equation Based on |x|

Since we have an absolute value, consider two cases. Case (i): \(x \geq 0\), where \(|x| = x\); the equation becomes \(a x^2 + bx + c = 0\). Case (ii): \(x < 0\), where \(|x| = -x\); the equation becomes \(a x^2 - bx + c = 0\).

Step 3: Analyze Case for \(x \geq 0\)

For \(x \geq 0\), solve \(a x^2 + bx + c = 0\). The discriminant \(\Delta_1 = b^2 - 4ac\). For real roots, \(\Delta_1\) must be non-negative, \(\Delta_1 \geq 0\).

Step 4: Analyze Case for \(x < 0\)

For \(x < 0\), solve \(a x^2 - bx + c = 0\). The discriminant \(\Delta_2 = b^2 - 4ac\) (same as the previous). For real roots, \(\Delta_2\) must be non-negative, \(\Delta_2 \geq 0\).

Step 5: Conclusion on Real Roots

In both cases, the discriminant does not change and must be non-negative for real roots. If \(\Delta_1 = \Delta_2 > 0\), each quadratic equation can have 2 distinct roots, leading to a total of 4 roots for both cases combined. Therefore, there can be 4 real roots as a possibility.

Key Concepts

Absolute Value Equations

When dealing with quadratic equations that involve absolute values, like \(a x^2 + b|x| + c = 0\), there are a few important considerations to make. The absolute value \(|x|\) introduces a major characteristic of the equation, as it can affect the algebraic behavior based on the sign of \(x\). To address this:

• Consider two main cases: when \(x \geq 0\) and when \(x < 0\).
• For \(x \geq 0\), \(|x| = x\) and the equation can be expressed as \(a x^2 + bx + c = 0\).
• For \(x < 0\), \(|x| = -x\) and the equation translates to \(a x^2 - bx + c = 0\).

This bifurcation essentially transforms one equation into two separate quadratic equations, each with its specific domain of definition determined by the sign of \(x\). Understanding these cases is crucial, as the absolute value functions differently depending on whether the variable inside it is positive or negative.

Real Roots

In quadratic equations, real roots are the solutions for which the quadratic equation holds true and they are real numbers. When solving an equation like \(a x^2 + b|x| + c = 0\), determining the number of real roots involves analyzing the two cases we've established.

• For \(x \geq 0\), solve the equation \(a x^2 + bx + c = 0\). The roots here are real if the discriminant (notated as \(\Delta_1\)) is non-negative.
• For \(x < 0\), solve \(a x^2 - bx + c = 0\). The roots are real if the discriminant (\(\Delta_2\)) is likewise non-negative.

If either or both of these equations have non-negative discriminants and produce real roots, these solutions contribute to the total number of real roots for the original absolute value problem. Depending on the scenario, this setup could yield different combinations of roots, which become apparent once the discriminants are evaluated.

Discriminants

The concept of the discriminant is pivotal in understanding the roots of quadratic equations. It is denoted by \(\Delta\) and typically calculated as \(b^2 - 4ac\). Here’s how it works:

• If \(\Delta > 0\), the quadratic equation has two distinct real roots.
• If \(\Delta = 0\), there is exactly one real root (a repeated root).
• If \(\Delta < 0\), the quadratic equation does not have any real roots (the roots are complex).

For our equation \(a x^2 + b|x| + c = 0\), the discriminant \(\Delta_1 = b^2 - 4ac\) is the same for both cases (\(x \geq 0\) and \(x < 0\)) as it does not involve the absolute value directly.Evaluating \(\Delta\) is critical because it determines whether real roots exist. In this situation, if both \(\Delta_1\) and \(\Delta_2\) are greater than zero, it can imply up to four distinct roots across the two cases. This reinforces the understanding that examining the discriminant gives us a powerful indicator of the potential real solutions in polynomial equations.