Problem 123 - A Complete Resource Book in Mathematics for JEE Main

Question

If the equation \(x^{2}+a^{2} x+b^{2}=0\) has two roots each of which exceeds a number \(c\), then (A) \(a^{4}>4 b^{2}\) (B) \(c^{2}+a^{2} c+b^{2}>0\) (C) \(-\frac{a^{2}}{2}>c\) (D) \(-\frac{a^{2}}{2}

Step-by-Step Solution

Answer

Option D is correct.

Step 1: Understand the Problem

The problem asks us to determine the condition under which the quadratic equation \(x^{2}+a^{2} x+b^{2}=0\) has roots that are greater than a constant \(c\). Each option provided gives a condition to satisfy.

Step 2: Use the Quadratic Formula

The quadratic roots for the equation \(x^{2}+a^{2}x+b^{2}=0\) can be found using the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]In this case, \(a = 1\), \(b = a^2\), and \(c = b^2\), giving us:\[x = \frac{-a^2 \pm \sqrt{(a^2)^2 - 4b^2}}{2}\]

Step 3: Find Conditions for Roots

For the roots to be real and greater than \(c\), the discriminant \((a^2)^2 - 4b^2\) must be non-negative:\[a^4 - 4b^2 \geq 0\] implying \[a^4 \geq 4b^2\]

Step 4: Analyze Roots

The greatest value among roots determined by the quadratic formula is \[x_1 = \frac{-a^2 + \sqrt{a^4 - 4b^2}}{2}\]For both roots to be greater than \(c\), this root must satisfy:\[-a^2 + \sqrt{a^4 - 4b^2} > 2c\] implying \[-a^2 > 2c\] or \(c < -\frac{a^2}{2}\).

Step 5: Validate Options

The condition from Step 4, specifically that \(c < -\frac{a^2}{2}\), corresponds to option D. This means the correct condition for roots exceeding \(c\) is option D.

Key Concepts

Roots of Equations

In order to find the roots of a quadratic equation, we often utilize the quadratic formula. The quadratic formula provides us a way to solve any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is given by:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula tells us the points where the quadratic equation equals zero, known as its roots. The \(\pm\) symbol indicates there are generally two roots, which may be real or complex depending on the discriminant (which we will discuss later).
Let's apply this to the equation mentioned: \(x^{2}+a^{2}x+b^{2}=0\). Here, \(a = 1\), \(b = a^2\), and \(c = b^2\). By plugging these into the quadratic formula, we attain our roots by evaluating:

• \(x = \frac{-a^2 \pm \sqrt{a^4 - 4b^2}}{2}\)

Evaluating these roots provides insight into where the graph of the quadratic will intersect the x-axis.
This information helps us in determining conditions, like ensuring roots are above a certain value \(c\).

Discriminant of a Quadratic Equation

The discriminant of a quadratic equation, expressed as \(b^2 - 4ac\), plays a crucial role in determining the nature and number of roots. For our equation, this becomes \(a^4 - 4b^2\). The discriminant helps us understand whether:

• The roots are real and distinct: If the discriminant is greater than zero.
• The roots are real and equal: If the discriminant is exactly zero.
• The roots are complex: If the discriminant is less than zero.

In the context of the problem, to ensure that our equation has real roots, we need the discriminant \(a^4 - 4b^2 \geq 0\). This ensures the square root in the quadratic formula is real, providing us with meaningful solutions.
By determining the sign and value of the discriminant, we align our expectations with the reality of the graph of the quadratic equation.

Inequality Solutions

Solving inequalities, especially those involving quadratic equations, involves ensuring that the solutions (i.e., roots) satisfy certain conditions. For the problem addressed, our task was to satisfy the inequality condition where all roots exceed the constant \(c\).
Once we find the greatest value of the roots using the quadratic formula,

• \(x_1 = \frac{-a^2 + \sqrt{a^4 - 4b^2}}{2}\)

We check the inequality condition \(-a^2 + \sqrt{a^4 - 4b^2} > 2c\). This derives to the condition \(c < -\frac{a^2}{2}\), meaning \(c\) should be less than half of the negative coefficient of \(x\).
This insight ensures when we solve inequalities in quadratic equations, we consider both algebraic transformations and logical deductions about where roots are located in relation to the number \(c\). Analyzing inequalities fundamentally allows us to draw conclusions about variable boundaries and constraints.