Problem 36 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(a\) and \(b\) are odd integers then \([x]^{2}+a[x]+b=0\) (where [ \(\cdot\) ] denotes greatest integer function) has (A) finite number of roots (B) infinite number of roots (C) no roots (D) None of these

Step-by-Step Solution

Answer

The equation has no roots.

Step 1: Understand the Problem

We are given a quadratic equation of the form \([x]^2 + a[x] + b = 0\), where \(a\) and \(b\) are odd integers, and \([\cdot]\) denotes the greatest integer function. We need to determine the number of roots the equation has.

Step 2: Analyze the Quadratic Equation

Rewrite the equation as \([x]^2 + a[x] + b = 0\). Here, \([x]\) represents the greatest integer less than or equal to \(x\), which means for any integer \(x = n\), \([x] = n\). Our task is to solve \(n^2 + an + b = 0\) for integer values of \(n\).

Step 3: Substitute Values and Solve

Since \(a\) and \(b\) are odd, substitute integer values for \(n\) in \(n^2 + an + b = 0\). Since \(a\) and \(b\) are odd, the quadratic can be rearranged to \(n(n + a) = -b\). Notice that the product \(n(n + a)\) must be even to satisfy an equality with an odd \(b\), which is impossible because \(ab\) should be odd, hence their product can never yield \(-b\).

Step 4: Conclusion of Root Analysis

Conclude from the analysis: since an odd product can never equal \(-b\) due to parity (even*even=even, odd*odd=odd, odd*even=odd etc.) constraints, the equation \(n^2 + an + b = 0\) has no integer solutions for \(n\). Therefore, \([x]^2 + a[x] + b = 0\) has no integer roots.

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation that generally takes the form \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants with \( a eq 0 \). The solutions to a quadratic equation, also known as roots, are the values of \( x \) that satisfy the equation.

• The equation can have zero, one, or two real roots.
• To solve a quadratic equation, you can use various methods, such as factoring, completing the square, or the quadratic formula.
• The discriminant \( b^2 - 4ac \) helps determine the number and type of roots.

In the context of the exercise, the quadratic equation involves the greatest integer function. This means we have an equation of the form \( [x]^2 + a[x] + b = 0 \), where solving for integer values becomes essential. This scenario introduces unique challenges because the relationship between the function's components can behave differently compared to standard quadratic equations.

Odd Integers

Odd integers are numbers that cannot be evenly divided by two. They have a remainder of one when divided by two.

• Examples of odd integers include \(-3, 1, 5, 7,\) and many more.
• An odd number can generally be represented in the form \( 2k + 1 \), where \( k \) is an integer.

The properties of odd integers play a significant role in determining solutions to certain types of equations. In this exercise, both \(a\) and \(b\) are odd integers, impacting the ability to find integer roots. The odd nature of these coefficients affects the parity of the quadratic expression \( n^2 + an + b \), leading us to conclude that no integer \( n \) will satisfy the equation.

Integer Solutions

Finding integer solutions to an equation means identifying integer values that make the equation true. When working with functions that involve the greatest integer function, identifying these solutions can be more complex.

• Integer solutions are whole numbers like \(-2, 0, 3\), etc.
• These solutions are particularly relevant in problems involving polynomials because not all polynomial equations have integer solutions.
• For a quadratic equation like this exercise's, if integer solutions exist, they can be found by substitution or derived methods.

The exercise's struggle to find integer solutions highlights how unique aspects of equations can lead to more abstract reasoning. The parity properties introduced through the odd coefficients \( a \) and \( b \) in the exercise make the existence of integer solutions improbable, thus yielding no roots for \( [x]^2 + a[x] + b = 0 \). Understanding the interplay between integer constraints and equation properties is crucial in such analyses.