Problem 151 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the difference between the roots of the equation \(x^{2}+a x+1=0\) is less than \(\sqrt{5}\), then the set of possible values of a is [2007] (A) \((-3,3)\) (B) \((-3, \infty)\) (C) \((3, \infty)\) (D) \((-\infty,-3)\)

Step-by-Step Solution

Answer

Option (A): The set of possible values for \(a\) is \((-3, 3)\).

Step 1: Understanding the Problem

Given the equation \(x^2 + ax + 1 = 0\), it is a quadratic equation in the form \(x^2 + bx + c = 0\) where \(b = a\) and \(c = 1\). The task is to find the values of \(a\) such that the difference between the roots is less than \(\sqrt{5}\).

Step 2: Recall the Discriminant and Roots Formula

The roots \(x_1\) and \(x_2\) of the quadratic equation \(ax^2 + bx + c = 0\) are given by the formula: \(x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). The difference between the roots can be calculated using the discriminant \(\Delta = b^2 - 4ac\) as \(|x_1 - x_2| = \frac{\sqrt{\Delta}}{|a|}\).

Step 3: Substitute Values into Discriminant Formula

In the given equation, \(b = a\) and \(c = 1\), so the discriminant \(\Delta\) is \(a^2 - 4\cdot1\cdot1 = a^2 - 4\). The difference between the roots \(|x_1 - x_2|\) is given by \(\frac{\sqrt{a^2 - 4}}{|1|} = \sqrt{a^2 - 4}\).

Step 4: Set Up the Inequality for Difference of Roots

We need the condition that the difference \(|x_1 - x_2| = \sqrt{a^2 - 4}\) is less than \(\sqrt{5}\). Therefore, we form the inequality: \(\sqrt{a^2 - 4} < \sqrt{5}\).

Step 5: Solve the Inequality

Square both sides to obtain \(a^2 - 4 < 5\). Simplifying further, we have \(a^2 < 9\). Solving this gives \(-3 < a < 3\).

Step 6: Identify the Appropriate Answer

The set of possible values for \(a\) is \((-3, 3)\), which corresponds to option (A).

Key Concepts

Understanding the Discriminant

The discriminant is a crucial concept when dealing with quadratic equations. It helps us determine the nature of the roots of these equations. The discriminant, denoted by \( \Delta \), is calculated using the formula \( \Delta = b^2 - 4ac \). Here, \( b \) and \( c \) are the coefficients of the terms in the standard form of a quadratic equation \( ax^2 + bx + c = 0 \).
\( \Delta \) tells us about the roots as follows:

• If \( \Delta > 0 \), there are two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root (also known as a repeated root).
• If \( \Delta < 0 \), there are no real roots; instead, the roots are complex.

The discriminant does not just indicate the type of roots. It is also used for finding specific details about the roots, such as their difference, as shown in the given exercise.

Exploring the Roots of Quadratic Equations

Roots of a quadratic equation are values of \( x \) that make the equation equal to zero. For an equation like \( ax^2 + bx + c = 0 \), the roots can be found using the quadratic formula:\[x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]The term inside the square root, \( b^2 - 4ac \), is the discriminant. This term is crucial for determining whether the roots are real or complex. In the given problem, since we are looking at the difference between the roots, we specifically deal with the absolute value of this difference:\[|x_1 - x_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}\]In our problem, since \( a = 1 \), the formula simplifies, making it easier to compare the roots' difference with the given condition of being less than \( \sqrt{5} \). Understanding the nature and calculation of roots allows us to solve inequalities involving them.

Solving Inequalities with Quadric Roots

Inequality solving is often needed when dealing with conditions on the roots of quadratic equations, such as in our exercise. When we set up the inequality for the difference between roots, it involves the discriminant:\[\sqrt{a^2 - 4} < \sqrt{5}\]To resolve it, we eliminate the square roots by squaring both sides:\[a^2 - 4 < 5\]This results in the inequality \( a^2 < 9 \). To solve this:

• We take the square root of both sides, resulting in \( -3 < a < 3 \).
• This gives us the interval for \( a \) which satisfies the initial condition given by the problem.

Solving inequalities is a powerful tool when dealing with mathematical problems involving constraints. The process of squaring both sides and analyzing the resulting algebraic conditions are essential steps in deriving the interval solution correctly.