Problem 82 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(a, b, c\) be real, if \(a x^{2}+b x+c=0\) has two real roots \(\alpha\) and \(\beta\), where \(\alpha<-1\) and \(\beta>1\) then \(1+\frac{c}{a}+\left|\frac{b}{a}\right|\) is (A) \(<0\) (B) \(\geq 0\) (C) \(\leq 0\) (D) None of these.

Step-by-Step Solution

Answer

The expression is less than 0, so answer (A): <0.

Step 1: Understand the given quadratic equation

The quadratic equation given is \( ax^2 + bx + c = 0 \) which has roots \( \alpha \) and \( \beta \). It is noted that \( \alpha < -1 \) and \( \beta > 1 \). We will use these root conditions to explore the expression \( 1 + \frac{c}{a} + \left|\frac{b}{a}\right| \).

Step 2: Use Vieta's formulas for roots

According to Vieta's formulas, for a quadratic equation \( ax^2 + bx + c = 0 \), the sum of roots \( \alpha + \beta \) is given by \( -\frac{b}{a} \), and the product of roots \( \alpha \cdot \beta \) is \( \frac{c}{a} \).

Step 3: Analyze the conditions of roots

Given \( \alpha < -1 \) and \( \beta > 1 \), then \( \alpha + \beta < -1 + 1 = 0 \). Thus, \( -\frac{b}{a} < 0 \), or \( \frac{b}{a} > 0 \). Also, \( \alpha \cdot \beta = \frac{c}{a} < -1 \times 1 = -1 \); hence, \( \frac{c}{a} < -1 \).

Step 4: Evaluate the expression

We need to consider the expression \( 1 + \frac{c}{a} + \left|\frac{b}{a}\right| \). From the analysis of roots, we know \( \frac{c}{a} < -1 \) and \( |\frac{b}{a}| = \frac{b}{a} > 0 \). Therefore, \( 1 + \frac{c}{a} < 1 - 1 = 0 \). Adding \(|\frac{b}{a}| > 0 \) to a negative result will keep the expression less than 0.

Key Concepts

Vieta's Formulas

Vieta's formulas offer an elegant way to swiftly connect the roots of a quadratic equation to its coefficients. When dealing with a quadratic equation in the form \( ax^2 + bx + c = 0 \), Vieta's formulas tell us that the sum of the roots \( \alpha + \beta = -\frac{b}{a} \) and the product \( \alpha \cdot \beta = \frac{c}{a} \).
These relationships are vital for simplifying complex expressions involving the roots without solving the quadratic equation for exact values. For example, if you know the coefficients \( a, b, \) and \( c \), you can directly find relationships between the roots and these coefficients. Instead of solving the quadratic equation, you can verify many properties of the roots by using these simple formulas.

• Sum of roots: \( \alpha + \beta = -\frac{b}{a} \)
• Product of roots: \( \alpha \cdot \beta = \frac{c}{a} \)

This can save time in mathematical proofs and provide deeper insights into polynomial functions.

Real Roots Conditions

The condition for real roots involves understanding the quadratic discriminant, given by the formula \( b^2 - 4ac \). For a quadratic equation \( ax^2 + bx + c = 0 \) to have real roots, the discriminant must satisfy \( b^2 - 4ac \geq 0 \). This ensures that the roots are real numbers and not complex.
In certain problems, knowing additional conditions such as \( \alpha < -1 \) and \( \beta > 1 \) can help refine our solutions. Such constraints help limit the values that the roots can take:
\( \bullet \) Thus, knowing \( \alpha + \beta < 0 \) allows us to deduce that \( \frac{b}{a} > 0 \).
\( \bullet \) Similarly, given that \( \alpha \cdot \beta < -1 \) implies \( \frac{c}{a} < -1 \), provides a bound to our values of interest.
Understanding these principles is crucial for analyzing quadratic expressions when faced with bounds or intervals for the roots.

Quadratic Inequalities

Quadratic inequalities deal with finding the range of values for variables that satisfy a quadratic equation. These are slightly more complex than linear inequalities because they can have two solutions or critical values, leading to a range of possible solutions. When structuring a calculation like \( 1 + \frac{c}{a} + \left| \frac{b}{a} \right| < 0 \), understanding inequalities is crucial.
Let's break down the approach:

• Assess whether each part of the quadratic is positive, negative, or zero.
• Consider the known inequalities; for example, knowing \( \frac{c}{a} < -1 \).
• Recognize the absolute value condition \( \left| \frac{b}{a} \right| = \frac{b}{a} > 0 \), which remains positive.

By properly combining these insights, you can determine the overall inequality of the expression. In this case, the operation indicated simplification and analysis showed that \( 1 + \frac{c}{a} + \left| \frac{b}{a} \right| < 0 \). Such evaluations play a significant role in analyzing real-world phenomena expressed through quadratic models.