Problem 65 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(a x^{2}+b x+c=0, a \neq 0, a, b, c \in R\) has distinct real roots in \((1,2)\) then \(a\) and \(5 a+2 b+c\) have (A) same sign (B) opposite sign (C) not determined (D) None of these

Step-by-Step Solution

Answer

The signs of \(a\) and \(5a + 2b + c\) are opposite (B).

Step 1: Identify the Conditions for Quadratic Roots

The quadratic equation \(ax^2 + bx + c = 0\) having distinct real roots implies that the discriminant must be greater than zero: \(b^2 - 4ac > 0\). Another given condition is that the roots of the equation lie in the interval \((1,2)\). This implies that for the roots \( \alpha \) and \( \beta \), \(1 < \alpha < \beta < 2\).

Step 2: Apply Vieta's Formulas for Root Conditions

Based on Vieta's formulas, for a quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots is \(-\frac{b}{a}\) and the product of the roots is \(\frac{c}{a}\). Therefore: \(1 < \frac{-b}{a} < 2\) and \(1 < \frac{c}{a} < 4\).

Step 3: Derive the Interval for Coefficient \(b\)

From the root sum condition \(\frac{-b}{a}\), multiply by \(-a\) (noting \(a eq 0\)): \[-a < b < -2a\].

Step 4: Determine \(c\) Based on Product of Roots

From the product of the roots \(\frac{c}{a} < 4\), it follows that \(c < 4a\).

Step 5: Translate Root Conditions to Conditions on \(5a + 2b + c\)

Starting with the inequalities: \(-a < b < -2a\) and \(c < 4a\). Calculate \(5a + 2b + c\). Substitute \(a < 0\) (for roots in the interval considered real) which causes \(b < -2a\) to result in \(5a + 2b + c < 0\) indicating an opposite sign to \(a\), which is negative.

Key Concepts

Discriminant

When solving quadratic equations, one key element to focus on is the discriminant. The discriminant is a part of the quadratic formula that dictates the nature of the roots.
The quadratic formula is given by:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]
The term inside the square root, \(b^2 - 4ac\), is what we call the discriminant.

• If \(b^2 - 4ac > 0\), the quadratic equation has two distinct real roots.
• If \(b^2 - 4ac = 0\), the equation has exactly one real root (also called a repeated root).
• If \(b^2 - 4ac < 0\), there are no real roots, only complex ones.

In the problem, we are given the discriminant condition \(b^2 - 4ac > 0\), ensuring distinct roots.

Real Roots

Real roots are the values of \(x\) that solve the quadratic equation and are real numbers. For this particular problem, the roots are explicitly mentioned to be distinct and real, and they lie within the interval \((1,2)\).
This is an important detail since it gives us further boundary conditions for the coefficients of the equation. Understanding real roots involves acknowledging their behavior on a graph of the quadratic function.
A quadratic equation has a parabolic graph and real roots represent the points where the parabola intersects the x-axis. More specifically:

• If the roots are distinct, the parabola will intersect the x-axis at two separate points.
• The fact that they lie between 1 and 2 tells us about the "spread" and "direction" of the parabola, indicating how the coefficients need to adjust to fit this shape.

Vieta's Formulas

Vieta's formulas provide a valuable relationship between the coefficients of a polynomial and its roots. For quadratic equations of the form \(ax^2 + bx + c = 0\):

• The sum of the roots \(\alpha + \beta\) is given by \(-\frac{b}{a}\).
• The product of the roots \(\alpha\beta\) is \(\frac{c}{a}\).

These are derived from expanding \((x-\alpha)(x-\beta)\) which equals \(x^2 - (\alpha + \beta)x + \alpha\beta\).
Incorporating these into the task:

• We know \(1 < \alpha + \beta < 2\).
• We also discover \(1 < \alpha\beta < 4\), which ties into the restrictions for coefficients \(b\) and \(c\).

Root Intervals

Understanding root intervals involves figuring out within what range the roots of a quadratic equation lie. Given that the real roots of the equation \(ax^2 + bx + c = 0\) lie in \((1, 2)\), we derive specific conditions for \(a\), \(b\), and \(c\).
This understanding comes from:

• The condition for the roots \(\alpha\) and \(\beta\) being between 1 and 2 is expressed as \( 1 < \alpha < \beta < 2\).
• By applying inequalities via Vieta's formulas, it leads us to the conditions \(-a < b < -2a\) and \(c < 4a\).

With these conditions, you can now evaluate the original equation in terms of signs:

• If \(a < 0\), this yields information on how \(5a + 2b + c\) behaves, leading to conclusions about their signs in determining the correct answer for the problem.