Problem 42 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\alpha\) and \(\beta(\alpha<\beta)\) be two different real roots of the equation \(a x^{2}+b x+c=0\), then (A) \(\alpha>-\frac{b}{2 a}\) (B) \(\beta<-\frac{b}{2 a}\) (C) \(\alpha<-\frac{b}{2 a}<\beta\) (D) \(\beta<-\frac{b}{2 a}<\alpha\)

Step-by-Step Solution

Answer

Option (C) \( \alpha < -\frac{b}{2a} < \beta \) is correct.

Step 1: Understand the properties of quadratic equations

A quadratic equation of the form \( ax^2 + bx + c = 0 \) has roots \( \alpha \) and \( \beta \). These roots are real and distinct if the discriminant \( b^2 - 4ac > 0 \). Additionally, the sum of the roots given by Vieta’s formulas is \( \alpha + \beta = -\frac{b}{a} \) and the product is \( \alpha\beta = \frac{c}{a} \). The vertex of the parabola is given by \( x = -\frac{b}{2a} \).

Step 2: Analyze the position of \(-\frac{b}{2a}\)

The vertex \( x = -\frac{b}{2a} \) is the midpoint of the roots when the roots are real and distinct. This implies \( \alpha < -\frac{b}{2a} < \beta \) since it represents the turning point of the parabola, which is symmetric concerning the line \( x = -\frac{b}{2a} \).

Step 3: Compare options with the derived inequality

The inequality from Step 2 \( \alpha < -\frac{b}{2a} < \beta \) directly corresponds to option (C). Thus, option (C) is the one that matches the condition derived for two distinct real roots and their respective positions with respect to \( -\frac{b}{2a} \).

Key Concepts

Distinct Real Roots

When dealing with quadratic equations of the form \( ax^2 + bx + c = 0 \), one interesting aspect is whether the roots of the equation are real and distinct. The nature of the roots is dictated by the discriminant, \( \Delta = b^2 - 4ac \).
For roots to be real and distinct, the discriminant must be greater than zero: \( \Delta > 0 \). This means:\
* The parabola associated with the quadratic function will intersect the x-axis at two different points.
* The solutions for \( x \), which are the points where the parabola crosses the x-axis, will be two different real numbers \( \alpha \) and \( \beta \).
Understanding if the roots are real and distinct helps in determining how many solutions a quadratic equation will have and how they behave.

Vertex of Parabola

The vertex of a parabola associated with a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula \( x_v = -\frac{b}{2a} \). This point is crucial when examining the graph of the quadratic function.
The vertex represents the parabola's turning point, where it switches directions. In a downward opening parabola (when \( a < 0 \)), this is the maximum point, and in an upward opening parabola (when \( a > 0 \)), this is the minimum point.

• If the parabola has real and distinct roots, the vertex is exactly between the roots.
• This means that if \( \alpha \) and \( \beta \) are the roots, the vertex line \( x = -\frac{b}{2a} \) divides the roots such that \( \alpha < -\frac{b}{2a} < \beta \).
• Thus, knowing the vertex can help in graph sketching and in finding the range of the function.

Always remember this point when graphing or solving problems, as it is a symmetric feature of all parabolas.

Vieta's Formulas

Vieta's Formulas offer a simple and fascinating way to relate the coefficients of a polynomial equation to the sum and product of its roots. For quadratics \( ax^2 + bx + c = 0 \) with roots \( \alpha \) and \( \beta \), Vieta's Formulas are:

• The sum of the roots: \( \alpha + \beta = -\frac{b}{a} \)
• The product of the roots: \( \alpha\beta = \frac{c}{a} \)

These formulas stem directly from the quadratic formula and the structure of polynomial division.
Using Vieta’s Formulas provides information about the roots without solving the quadratic formally. For example:

• The sum formula is particularly useful for finding intermediate values or symmetry of a parabola.
• The product formula gives an insight into the possible signs of the roots relative to the constant term \( c \).

They empower you to handle quadratic equations with finesse, facilitating both problem-solving and understanding of the relationship between the polynomial's coefficients and its roots.