Problem 112 - A Complete Resource Book in Mathematics for JEE Main

Question

The roots of the equation \(a x^{2}+b x+c=0\), where \(a \in R^{+}\), are two consecutive odd positive integers, then (A) \(|b| \leq 4 a\) (B) \(|b| \geq 4 a\) (C) \(|b|=2 a\) (D) None of these

Step-by-Step Solution

Answer

(B) \(|b| \geq 4a\)

Step 1: Define the roots

Let the roots of the quadratic equation be two consecutive odd positive integers. Denote these roots by \( n \) and \( n+2 \), where \( n \) is an odd positive integer.

Step 2: Use Vieta's formulas

According to Vieta's formulas, the sum of the roots \( n + (n+2) \) is equal to \(-\frac{b}{a}\). Thus, \( b = -a(2n + 2) \).

Step 3: Express \( b \) in terms of \( a \)

Simplifying, we have \( b = -2a(n+1) \). The absolute value \(|b|\) is therefore \(2a(n+1)\).

Step 4: Compare \(|b|\) with \(4a\)

To find the relationship between \(|b|\) and \(4a\), compare \(2a(n+1)\) with \(4a\). We see that \(2a(n+1) = 4a\) when \(n+1 = 2\), thus \(|b| = 4a\) is possible.

Step 5: Determine the answer to the problem

Since there can be values of \(n\) such that \(n+1 = 2\) does not correspond to any valid integer that fits our original assumption of consecutive odd integers, revisit our logic. If \(n=1\), the roots would be 1 and 3, making \( b = -8a \) which doesn't fit exactly. The right mathematical condition is that \(|b| \geq 4a\) after comparing all cases.

Key Concepts

Consecutive Odd Integers

In mathematics, understanding the concept of consecutive odd integers is crucial, especially when dealing with quadratic equations. Consecutive odd integers are numbers that follow each other in order, with a difference of two between them—like 1, 3, 5, and so on. This sequence is clear-cut as every odd number is separated by exactly two from its neighbor.

When used in quadratic equations, such as finding the roots of a polynomial, these integers can help in creating specific conditions or constraints to find particular solutions. For instance, if we're told that the roots of a given quadratic equation are consecutive odd integers, it means if one root is denoted as \( n \), the next will be \( n+2 \). This understanding can help simplify solving the equation or applying other mathematical formulas like Vieta's formulas to reveal more about the relationship between the coefficients and roots.

Vieta's Formulas

Named after the French mathematician François Viète, Vieta's formulas are powerful tools in algebra. They relate the coefficients of a polynomial to the sums and products of its roots. For a quadratic equation of the form \( ax^2 + bx + c = 0 \), Vieta's formulas tell us that the sum of the roots \( \alpha \) and \( \beta \) is \( -\frac{b}{a} \), and the product of the roots is \( \frac{c}{a} \).

In the context of our exercise, if the roots are \( n \) and \( n+2 \), then using Vieta's first formula, the sum \( n + (n+2) = 2n + 2 \) equals \( -\frac{b}{a} \). This relationship allows us to express \( b \) in terms of \( a \) and \( n \) as \( b = -2a(n+1) \).

• Sum of Roots: Relates directly to \( b \)
• Product of Roots: Can help find \( c \) when solving quadratic equations

Understanding these links is crucial in unraveling the properties of quadratic equations and finding simpler paths to their solutions.

Absolute Value of Coefficients

The absolute value of a number represents its distance from zero, disregarding its sign. In algebra, when dealing with coefficients of an equation, their absolute values can indicate bounds, relationships, or conditions critical to solving the equation.

For example, in the quadratic equation \( ax^2 + bx + c = 0 \) with specific roots like consecutive odd integers, the absolute value \(|b|\) allows comparison to other expressions. In our problem, we needed to compare \(|b|\) to \(4a\) to determine what must be true about \( b \) for the roots to be consecutive integers.

• Computation: \(|b| = 2a(n+1)\)
• Comparison: Helps establish conditions like \(|b| \geq 4a\)
• Application: Aids in narrowing down viable solutions based on the absolute values

Understanding absolute values in this context is critical, as it helps analyze and solve equations with specific conditions or constraints.