Problem 112 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the equation \(a x^{2}+b x+c=0\) has two distinct positive roots, then the equation \(a x^{2}+(b+6 a) x+\) \((c+3 b)=0\) has (A) two positive roots (B) exactly one positive root (C) at least one positive root (D) no positive root

Step-by-Step Solution

Answer

(C) At least one positive root.

Step 1: Identify Conditions for Two Distinct Positive Roots

For the quadratic equation \(a x^2 + b x + c = 0\) to have two distinct positive roots, three conditions must be satisfied: 1. The discriminant \(D = b^2 - 4ac\) must be positive.2. The leading coefficient \(a\) must be greater than zero.3. The sum of the roots \(-\frac{b}{a}\) must be positive, which implies \( b < 0\).

Step 2: Analyze Modified Quadratic Equation

Consider the equation \(a x^2 + (b + 6a)x + (c + 3b) = 0\). Our aim is to determine the conditions under which this equation has positive roots by analyzing its coefficients and properties relative to the original equation.

Step 3: Determine Discriminant of Modified Equation

The discriminant of the modified equation is \((b + 6a)^2 - 4a(c + 3b)\). Simplifying, we have:\[(b + 6a)^2 - 4a(c + 3b) = b^2 + 12ab + 36a^2 - 4ac - 12ab = b^2 - 4ac + 36a^2\]Given the original condition \(b^2 - 4ac > 0\), add \(36a^2\) to it. This increases the positive nature of the discriminant.

Step 4: Check Sum of Roots for Modified Equation

The sum of the roots of the equation \(a x^2 + (b + 6a)x + (c + 3b) = 0\) is given by \(-\frac{b + 6a}{a}\). Since \(b < 0\) and \(a > 0\), \(b + 6a\) is likely to be negative, making the sum of the roots positive, confirming at least one positive root.

Step 5: Conclude Based on Analysis

Since the modified discriminant \((b^2 + 36a^2 - 4ac)\) is positive, the equation has two roots.With negative \((b + 6a)\), the sum of the roots is positive, ensuring at least one positive real root.

Key Concepts

Discriminant

In the world of quadratic equations, the discriminant plays a key role in determining the nature and type of roots. The discriminant is derived from the quadratic formula: given a quadratic equation of the form \(ax^2 + bx + c = 0\), it is represented by \(D = b^2 - 4ac\). This value tells us much about the roots:

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), there is exactly one real root (a repeated root).
• If \(D < 0\), the roots are complex and not real.

Understanding the discriminant helps in predicting whether a quadratic equation will have distinct real roots or not, which is critical in solving the equation effectively, as demonstrated in the exercise.

Roots of Quadratic Equation

To understand the roots of a quadratic equation, it’s imperative to recognize how they are derived and what they signify in any given quadratic form. Quadratic equations generally have two roots, which could be real or complex. These roots are solutions to the equation \(ax^2 + bx + c = 0\). The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) is used to compute these roots.The nature of the roots can be thoroughly understood by:

• Evaluating the Discriminant (\(D = b^2 - 4ac\)): As mentioned, a positive discriminant indicates two distinct real roots.
• Considering the sum and product of the roots: The sum of the roots is \(-\frac{b}{a}\), and the product of the roots is \(\frac{c}{a}\). These are essential in deriving relationships between the coefficients and roots.

In the context of the exercise, knowing that the original equation has two distinct positive roots ensures the roots are real and positive, leading us to evaluate if subsequent changes to the equation affect these properties.

Positive Roots

Positive roots in a quadratic equation imply that both solutions to the equation are greater than zero. Identifying when roots are positive involves verifying certain conditions in the quadratic form \(ax^2 + bx + c = 0\).Key factors for positive roots include:

• The sum of the roots \(-\frac{b}{a}\) should be positive, indicating that \(b\) must be negative when \(a > 0\).
• Since one condition for positivity is positivity of the product of roots \(\frac{c}{a} > 0\), it means \(c\) must be positive when \(a > 0\).

In the exercise, the transformation of the quadratic equation introduces new coefficients. It is crucial to reassess if these transformations still satisfy the conditions for positive roots. By examining how these adjustments influence the discriminant and the sum of the roots, you can determine that at least one root remains positive, as confirmed by our analysis.