Problem 125 - A Complete Resource Book in Mathematics for JEE Main

Question

If the equation \(x^{2}+(a-b) x-a-b+1=0\), where \(a\), \(b \in \mathrm{R}\), has unequal real roots for all \(b \in R\), then (A) \(a<0\) (B) \(a>0\) (C) \(a>1\) (D) \(a<1\)

Step-by-Step Solution

Answer

The correct answer is (C) \(a > 1\).

Step 1: Understand the nature of the roots

For the quadratic equation \(x^2 + (a-b)x - a - b + 1 = 0\) to have unequal real roots, the discriminant must be positive. The discriminant \(\Delta\) for a quadratic equation \(Ax^2+Bx+C=0\) is given by \(\Delta = B^2 - 4AC\). Substitute the equation parameters to find the discriminant.

Step 2: Calculate the Discriminant

Substitute \(A=1\), \(B=(a-b)\), and \(C=(-a-b+1)\) into the discriminant formula: \(\Delta = (a-b)^2 - 4 \cdot 1 \cdot (-a-b+1)\). Thus, the discriminant is \(\Delta = (a-b)^2 + 4(a+b-1)\).

Step 3: Ensure Discriminant is Positive for All b

For all \(b \in \mathbb{R}\), \(\Delta\) must be greater than zero. \((a-b)^2 + 4(a+b-1) > 0\). Rearrange the equation: \((a-b)^2 + 4a + 4b - 4 > 0\). Simplify it to \((a-b)^2 + 4a + 4b > 4\).

Step 4: Simplify and Analyze Critical Values

The discriminant formula \((a-b)^2 + 4a + 4b > 4\) must hold for all \(b\). Utilize 'worst case' by finding when \((a-b)^2\) is minimized, which happens at \(b = a\). Substitute to check: \(4a > 4\). Thus, solve to get \(a > 1\).

Step 5: Conclude the Analysis

Given the requirement is that \(\Delta > 0\) for all \(b\), and solving gives only \(a > 1\), then \(a\) must indeed be greater than 1 to satisfy the condition for all \(b\) values.

Key Concepts

Discriminant

The discriminant is a key feature in solving quadratic equations. It helps us determine the nature of the roots of the equation. For any quadratic equation of the form \( Ax^2 + Bx + C = 0 \), the discriminant \( \, \Delta \) is calculated using the formula \( B^2 - 4AC \). This small component plays a big role in identifying the type of roots present in the equation.

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), the equation has exactly one distinct real root, or a repeated root.
• If \( \Delta < 0 \), the equation has complex roots.

When dealing with questions like the one provided, ensuring that the discriminant remains positive, regardless of the values of other variables, is essential. This guarantees that the quadratic equation always yields real and unequal roots, showcasing the power and necessity of the discriminant in solving quadratic equations.

Real Roots

Real roots of a quadratic equation are values of \( x \) that satisfy the equation \( Ax^2 + Bx + C = 0 \) while being real numbers. Real roots can either be distinct, meaning they are two different numbers, or they can be identical, which are often referred to as repeated roots.

For an equation to have real roots, the discriminant \( \Delta \) must be zero or positive. However, to have distinct roots, \( \Delta \) should strictly be greater than zero. This means that the calculation of the discriminant from the original equation's coefficients is crucial in finding this. Understanding the nature of real roots helps in grasping solutions to the equation, making the process of graphing or predicting behavior of the equation's curve easier for students.

In scenarios like the exercise provided, skewing the coefficients such that \( \Delta \) remains positive allows for maintaining real, distinct roots regardless of changes in variable values, especially when dealing with inequalities.

Inequality in Quadratic Equations

Inequality in quadratic equations involves analyzing the solutions or values of \( x \) for which a quadratic expression maintains its inequality. In the context of the provided exercise, an inequality is formed from the discriminant \( \Delta \) to ensure it is always positive, as prescribed by the condition for having unequal, real roots.

The key here is to manipulate the inequality \( (a-b)^2 + 4(a+b-1) > 0 \) to find the feasible range of the parameter \( a \), irrespective of the value of \( b \). Simplifying inequalities like this is crucial:

• Plug in specific values or scenarios, like setting \( b = a \), to test the worst-case or boundary values.
• Analyze how changes in one variable affect the inequality.

By doing such exercises, students practice critical thinking necessary to solve not just specific quadratic problems but a wide array of mathematical problems where inequalities dictate conditions for solutions.