Problem 98 - A Complete Resource Book in Mathematics for JEE Main

Question

If both the roots of the quadratic equation \(x^{2}-2 k x+\) \(k^{2}+k-5=0\) are less than 5, then \(k\) lies in the interval (A) \((-\infty, 4)\) (B) \([4,5]\) (C) \((5,6]\) (D) \((6, \infty)\)

Step-by-Step Solution

Answer

k lies in the interval (6, ∞).

Step 1: Understand the Problem

We are given a quadratic equation \(x^2 - 2kx + k^2 + k - 5 = 0\) and need to determine the interval of \(k\) such that both roots are less than 5.

Step 2: Use the Condition for Roots Less than a Number

For both roots of the quadratic equation to be less than 5, the vertex of the parabola (which opens upwards) must be less than 5. This implies that \(f(5) > 0\), where \(f(x) = x^2 - 2kx + k^2 + k - 5\).

Key Concepts

Roots of Quadratic Equations

Quadratic equations are polynomial equations of degree two and are generally expressed in the standard form: \( ax^2 + bx + c = 0 \). The solutions to this equation, known as its roots, can be found using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). The term under the square root, \( b^2 - 4ac \), is called the discriminant and plays a critical role in determining the nature of the roots.

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If it is zero, there are exactly two real and identical roots.
• If the discriminant is negative, the roots are complex and not real numbers.

Understanding the roots is key, as they are the \(x\)-values where the quadratic function intersects the \(x\)-axis. In practical scenarios, we often need additional conditions such as the roots being less or more than a certain number, which demands careful analysis of the function's behavior at these points.

Inequalities in Quadratics

Inequalities involving quadratics are handled by analyzing the sign of the quadratic expression in question over different intervals. Unlike linear inequalities, where the sign changes at zero, for quadratic inequalities, the sign can change at the roots of the equation or be dictated by the parabola's shape itself.
To solve them, it's often helpful to:

• First, find the roots of the quadratic equation by setting it to zero \((ax^2 + bx + c = 0)\).
• Determine where the quadratic is greater than or less than zero.
• Use test points or sketch the parabola to figure out the interval over which the inequality holds.

Analyzing inequalities in quadratics involves not just finding roots, but also determining the nature and interval of the solutions. This can involve checking conditions such as "less than a certain value" or "greater than a particular range" to determine valid \(k\)-values in more advanced problems.

Parabola Vertex

The vertex of a parabola gives important information regarding its minimum or maximum point. For parabolas in the form \(ax^2 + bx + c\), the vertex \((h, k)\) is found using the formula \(h = \frac{-b}{2a} \) and substituting \(h\) back into the function to find \(k\).
In quadratic equations, particularly where conditions involve values being less than a certain number, the vertex plays a critical role. The vertex corresponds to the point on the \(y\)-axis where the parabola changes direction. For the inequality problems, understanding the vertex's position helps determine if the parabola's minimum or maximum satisfies given conditions.
For example, if both roots need to be less than a specific value, the vertex also must lie below this transversal line. This means evaluating the vertex in relation to conditions helps set boundaries on parameters like \(k\).

Solving Quadratic Inequalities

Solving quadratic inequalities involves combining understanding the roots, vertex, and overall behavior of the quadratic. Unlike quadratic equations with exact roots, inequalities require knowing where the parabola is positioned above or below the \(x\)-axis--or a specific line like \(x = 5\) in this case.
The solution typically involves these steps:

• Find the vertex and determine its coordinates.
• Analyze where the parabola opens—either upwards or downwards.
• Check the function value at critical points such as the set inequality number, here \(x = 5\).
• Evaluate and solve the resulting inequality, \(f(5) > 0\), to find the permissible values of \(k\).

This approach helps ensure that all conditions around quadratic inequalities are met, providing a comprehensive understanding of the inequality situation. They help us pinpoint intervals where parameters like \(k\) lie, effectively meeting all given conditions of the problem.