Problem 10 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If the root of \(x^{2}-b x+c=0\) are two consecutive integers, then \(b^{2}-4 c\) is (a) 1 (b) 2 (c) 3 (d) 4

Step-by-Step Solution

Answer

The correct answer is (a) 1.

Step 1: Understanding the Nature of Roots

Since the roots of the quadratic equation are two consecutive integers, let's assume that the roots are \( n \) and \( n+1 \). In a quadratic equation \( ax^2 + bx + c = 0 \) with roots \( p \) and \( q \), the sum of the roots is given by \( p + q = -\frac{b}{a} \) and the product of the roots is \( pq = \frac{c}{a} \). For our equation \( x^2 - bx + c = 0 \), \( a = 1 \).

Step 2: Set Up Equations Based on Root Properties

From the assumption about the roots, we use the sum \( n + (n+1) = 2n + 1 = b \). For the product of the roots: \( n(n+1) = n^2 + n = c \). This gives us two key relationships: \( b = 2n + 1 \) and \( c = n^2 + n \).

Step 3: Expressing the Discriminant

The discriminant of a quadratic equation is given by \( b^2 - 4ac \). Since \( a = 1 \), we need to compute \( b^2 - 4c \). Substitute \( b = 2n + 1 \) and \( c = n^2 + n \) into this expression to find its value.

Step 4: Calculate the Discriminant

Substitute the expressions for \( b \) and \( c \) into the discriminant formula:\[b^2 - 4c = (2n + 1)^2 - 4(n^2 + n) = 4n^2 + 4n + 1 - 4n^2 - 4n.\]Simplifying gives us:\[b^2 - 4c = 1.\]

Step 5: Determine the Correct Option

From the calculated discriminant \( b^2 - 4c = 1 \), we find that the correct choice is option (a).

Key Concepts

Roots of Equations

In algebra, every quadratic equation of the form \(ax^2 + bx + c = 0\) has roots. The roots are the solutions to the equation, where the equation equals zero. These roots can be real or complex, and they can also be the same or different. In simple terms, finding the roots of a quadratic equation is identifying the points where the graph of the equation intersects the x-axis.

Consider the quadratic equation \(x^2 - bx + c = 0\). If the roots are given as \(p\) and \(q\), then by Vieta's formulas:

• The sum of the roots, \(p + q\), is equal to \(-\frac{b}{a}\).
• The product of the roots, \(pq\), equals \(\frac{c}{a}\).

This relationship is useful as it allows us to find or check the values of \(b\) and \(c\) when the roots are known, or vice versa. It is particularly helpful when the roots are whole numbers or satisfy a certain condition, like being consecutive integers.

Discriminant

The discriminant is a key concept in understanding the nature of the roots of a quadratic equation. It is part of the quadratic formula and is found in the expression \(b^2 - 4ac\). The value of the discriminant can tell us several things about the roots:

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, there is exactly one real root, known as a repeated or double root.
• If the discriminant is negative, the roots are complex and not real.

In our given problem, the equation \(x^2 - bx + c = 0\) simplifies with known values for \(a = 1\), allowing easy calculation of the discriminant. By substituting the appropriate values into the formula, we determined that \(b^2 - 4c = 1\), which indicates that the roots are two consecutive integers.

Consecutive Integers

Consecutive integers are numbers that follow each other in order without any gaps. They are integers like 7 and 8, or 13 and 14. When a quadratic equation has consecutive integer roots, it means that if one root is \(n\), the other root will be \(n + 1\).

This property is useful because it simplifies calculations. In this exercise, by expressing the roots as \(n\) and \(n+1\), it becomes straightforward to use them in Vieta's formulas to find specific coefficients of the quadratic equation. By calculating the sum and product of these roots, one can derive other needed values such as \(b\) and \(c\). This method greatly aids in solving quadratic equations with consecutive integer roots.

Sum and Product of Roots

The sum and product of the roots are derived from the coefficients of the quadratic equation \(ax^2 + bx + c = 0\). Specifically:

• The sum of the roots \(p + q\) is \(-\frac{b}{a}\).
• The product of the roots \(pq\) is \(\frac{c}{a}\).

These formulas help when you know the roots and want to find the equation, or when you know the equation and want a relationship between its roots. In this exercise, the roots \(n\) and \(n+1\) were used to create expressions for \(b = 2n + 1\) (sum) and \(c = n^2 + n\) (product). This step was pivotal to find that \(b^2 - 4c = 1\) thus ensuring the correct choice was made.

Understanding how sums and products relate to roots provides a deeper grasp for solving quadratic equations efficiently.