Problem 5 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\alpha\) and \(\beta(\alpha<\beta)\), are the roots of the equation \(x^{2}+\) \(b x+c=0\), where \(c<0

Step-by-Step Solution

Answer

Option (D): \(\alpha < 0 < |\alpha| < \beta\) is correct.

Step 1: Understand Vieta's Formulas

For any quadratic equation \(ax^2 + bx + c = 0\), where \(a=1\) in this case, the sum of the roots \((\alpha + \beta)\) is equal to \(-b/a = -b\), and the product of the roots \(\alpha \beta\) is equal to \(c/a = c\). Here, \(b > 0\) and \(c < 0\).

Step 2: Analyze the Quadrant for Roots

Given \(c < 0\), this means that \(\alpha \beta < 0\). The product of the roots being negative implies that one root is positive and the other is negative. Further, \(\alpha < \beta\) suggests \(\alpha < 0 < \beta\).

Step 3: Check the Order and Magnitude of Roots

We know \(\alpha < \beta\) and \(\alpha < 0 < \beta\). Triangle inequality for roots in this form indicates that one of the roots is opposite in sign to the sum of roots \(\alpha + \beta = -b < 0\), meaning \(\beta\) must be greater in magnitude than \(\alpha\) but opposite in terms. This results in \(\alpha < 0 < |\alpha| < \beta\).

Step 4: Select the Correct Option

From Steps 2 and 3, we can see that the condition \(\alpha < 0 < |\alpha| < \beta\) is true. Thus, option (D) is correct.

Key Concepts

Vieta's Formulas

Vieta's Formulas offer a fundamental tool in understanding the relationship between the coefficients and roots of a quadratic equation. For a quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a\) is the leading coefficient, Vieta's Formulas tell us two key relationships:

• The sum of the roots, \(\alpha + \beta\), is equal to \(-\frac{b}{a}\).
• The product of the roots, \(\alpha \beta\), is equal to \(\frac{c}{a}\).

For the specific case when \(a = 1\) in our equation, these formulas become even simpler:

• \(\alpha + \beta = -b\)
• \(\alpha \beta = c\)

Given that \(b > 0\) and \(c < 0\), these relationships provide crucial insights into the nature of the roots. Specifically, the sum \(\alpha + \beta = -b\) is negative, indicating that the roots, in combination, result in a negative total value.

The product \(\alpha \beta = c\) being negative reveals that one root is positive, and the other is negative—a critical observation for analyzing root behavior in our problem.

Roots of Quadratic Equation

The roots of a quadratic equation, often symbolized as \(\alpha\) and \(\beta\), can reveal much about the equation's characteristics when analyzed cleverly. In our case, with the quadratic equation \(x^2 + bx + c = 0\), where \(b > 0\) and \(c < 0\), we have a clear path to understanding the root nature.

Given that \(\alpha \beta = c < 0\), one root must be positive and the other negative. Thus, we can assume \(\alpha < 0\) and \(\beta > 0\). Consequently, our task becomes ensuring the order \(\alpha < \beta\) is consistent with other given conditions.

Such observations are essential for determining the exact real-number values of \(\alpha\) and \(\beta\), which can only be done if additional numerical conditions are given. However, understanding the interaction between sum, product, and individual signs is crucial to picking apart potential options or solutions.

Signs of Roots

The signs of the roots in quadratic equations often determine the solutions' qualitative behavior and can be deduced from the product \(\alpha \beta\) and sum \(\alpha + \beta\). In this scenario, with \(\alpha \beta < 0\), it's clear one root is positive, and the other is negative. This finding highlights a key feature of root signs where opposites attract in terms of product negativity.

Analyzing signs:

• The requirement \(\alpha < \beta\) implies sequence order but is further enforced by contextually understanding that \(\alpha\) must be the negative root and \(\beta\) the positive one because of the nature of \(b\) and \(c\).
• \(\alpha < 0\), and since \(\alpha + \beta = -b < 0\), it means \(\beta\) is larger in magnitude than \(\alpha\).

This analytical observation ensures the root arrangement of \(\alpha < 0 < |\alpha| < \beta\) logically follows the expected order given the conditions, and confirms the selection of the correct option in the exercise.