Problem 147 - A Complete Resource Book in Mathematics for JEE Main

Question

If roots of the equation \(x^{2}-b x+c=0\) be two consectutive integers, then \(b^{2}-4 c\) equals \(\quad\) [2005] (A) \(-2\) (B) 3 (C) 2 (D) 1

Step-by-Step Solution

Answer

The correct answer is (D) 1.

Step 1: Recognize the Nature of Roots

The problem states that the two roots are consecutive integers. Let's denote these roots as \(n\) and \(n+1\).

Step 2: Write the Sum and Product of Roots

According to Vieta's formulas, for a quadratic equation \(x^2 - bx + c = 0\), the sum of the roots \(n + (n+1)\) is equal to \(b\). Therefore, \(b = 2n + 1\). Similarly, the product of the roots \(n(n+1)\) is equal to \(c\). Thus, \(c = n(n+1)\).

Step 3: Calculate the Difference \(b^2 - 4c\)

Substitute \(b = 2n + 1\) and \(c = n(n+1)\) into the expression \(b^2 - 4c\):\[b^2 - 4c = (2n + 1)^2 - 4(n(n+1))\]Expand and simplify:\[(2n + 1)^2 = 4n^2 + 4n + 1\]\[4(n(n+1)) = 4n^2 + 4n\]The expression becomes:\[4n^2 + 4n + 1 - 4n^2 - 4n = 1\]

Step 4: Conclude the Solution

Since all terms cancel out except for \(1\), we have calculated that \(b^2 - 4c = 1\).

Key Concepts

Quadratic Equation

A quadratic equation is a second-order polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants and \( a eq 0 \). In this context, the quadratic equation is given as \( x^2 - bx + c = 0 \). Quadratic equations often have two solutions or roots, which can either be real or complex numbers.
To find the roots of a quadratic equation, we can use the quadratic formula:

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

The term \( b^2 - 4ac \) is called the discriminant and it determines the nature of the roots. In our exercise, it's important because we're specifically working with the concept of roots being consecutive integers.

Vieta's Formulas

Vieta's formulas are a valuable tool when dealing with quadratic equations. They relate the coefficients of the equation to sums and products of its roots, without actually needing to solve the equation.
For a quadratic equation of the form \( x^2 - bx + c = 0 \), Vieta's formulas provide:

• The sum of the roots \( (r_1 + r_2) = b \)
• The product of the roots \( r_1 \cdot r_2 = c \)

In our exercise, we apply Vieta's formulas by considering the roots as consecutive integers \( n \) and \( n+1 \). Therefore, their sum is \( n + (n+1) = 2n + 1 \), which equals \( b \), and their product \( n(n+1) = n^2 + n \), which equals \( c \). This insight allows us to express \( b \) and \( c \) in terms of \( n \), leading to the simplification of the expression \( b^2 - 4c \).

Roots and Coefficients Relationship

Understanding the relationship between the roots of a polynomial and its coefficients is crucial for solving and simplifying such problems.
In a quadratic equation such as \( x^2 - bx + c = 0 \), the coefficients \( b \) and \( c \) are directly related to its roots via Vieta's formulas. This means:

• The sum of the roots is equal to the coefficient \( b \), allowing us to identify patterns like consecutive integers forming the roots.
• The product of the roots gives us the constant term \( c \).

In this exercise, recognizing that the roots are consecutive integers helps simplify the existing equation. This particular relationship enables us to substitute \( n \) into expressions for \( b \) and \( c \), simplifying \( b^2 - 4c \) down to 1. It beautifully demonstrates the power of using algebraic structure and relationships to solve problems efficiently.