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

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

(D) \(\alpha < 0 < |\alpha| < \beta\)

Step 1: Identify Given Conditions

We are given that \(\alpha\) and \(\beta\) are roots of the equation \(x^2 + bx + c = 0\), with conditions \(c < 0 < b\).

Step 2: Use Vieta's Formulas

According to Vieta's formulas, the sum of the roots \(\alpha + \beta = -b\), and the product of the roots \(\alpha\beta = c\).

Step 3: Analyze the Sum of Roots

Given that \(-b < 0\), it follows that \(\alpha + \beta < 0\). Therefore, one root is negative, and the sum of the roots is negative.

Step 4: Analyze the Product of Roots

Given \(c < 0\), it follows that \(\alpha\beta < 0\). This implies that one root is positive and the other is negative.

Step 5: Determine Possible Order of Roots

Combining facts: \(\alpha + \beta < 0\) and \(\alpha\beta < 0\), we find \(\alpha < 0 < \beta\), and for the sum \(\alpha + \beta < 0\), it must be that \(|\alpha| > \beta\).

Step 6: Select Correct Answer

The roots must satisfy \(\alpha < 0 < |\alpha| < \beta\), so the correct answer is option (D).

Key Concepts

Vieta's Formulas

Vieta's formulas are a set of mathematical relationships that relate the coefficients of a polynomial with sums and products of its roots. For a quadratic equation in the form of \( ax^2 + bx + c = 0 \), they provide us with two crucial relationships:

• The sum of the roots: \( \alpha + \beta = -\frac{b}{a} \)
• The product of the roots: \( \alpha\beta = \frac{c}{a} \)

These formulas essentially tell us how the roots \( \alpha \) and \( \beta \) are intertwined with the coefficients \( b \) and \( c \) of the equation. It’s like connecting the dots between the roots and the equation itself.
In the problem at hand, the equation is \( x^2 + bx + c = 0 \), which implies \( a = 1 \). So, specifically for this equation:

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

With \( c < 0 < b \), these formulas help us to further understand the nature and order of the roots.

Roots of Quadratic Equation

A quadratic equation typically looks like \( ax^2 + bx + c = 0 \). The solutions to this equation are its roots, \( \alpha \) and \( \beta \). These roots can be found using various methods like factoring, completing the square, or the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\]The roots tell us where the quadratic function will cross the x-axis. In graphical terms, they are the x-coordinates of the points where the parabola associated with the quadratic equation intersects the x-axis.
When solving such equations, knowing whether the roots are real, imaginary, or complex is crucial. This decision is based on the discriminant \( b^2 - 4ac \):

• If \( b^2 - 4ac > 0 \), the roots are real and distinct.
• If \( b^2 - 4ac = 0 \), the roots are real and repeated.
• If \( b^2 - 4ac < 0 \), the roots are complex and imaginary.

In this exercise, we know that the roots are real and of opposite signs because the product \( \alpha\beta = c < 0 \).

Relationships between Roots and Coefficients

The relationship between the roots and coefficients of a quadratic equation is a fundamental aspect that helps predict the behavior of the polynomial function without explicitly calculating the roots. When we say \( x^2 + bx + c = 0 \), Vieta's relationships—\( \alpha + \beta = -b \) and \( \alpha\beta = c \)—play a central role.
Given \( c < 0 \) and \( b > 0 \), we determine:

• The sum \( \alpha + \beta = -b \) is negative, suggesting that the larger magnitude root is negative.
• The product \( \alpha \beta = c < 0 \) means one root is negative, and the other is positive.

Thus, these conditions—when pieced together like a puzzle—indicate how the roots \( \alpha \) and \( \beta \) are structured around the linear (\( b \)) and constant (\( c \)) coefficients. In the context of the problem, these relationships reveal that \( \alpha < 0 < \beta \) and also \( |\alpha| > \beta \), which ultimately confirms option (D) as the correct answer.