Problem 15 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Let \(a, b, c\) be real numbers with \(a \neq 0\) and let \(\alpha, \beta\) be the roots of the equation \(a x^{2}+b x+c=0 .\) Express the roots of \(a^{3} x^{2}+a b c x+c^{3}=0\) in terms of \(\alpha, \beta\).

Step-by-Step Solution

Answer

The roots of the equation \(a^3x^2 + abc x + c^3=0\) can be expressed in terms of the roots of the equation \(ax^2 + bx + c=0\) as: \(\gamma = \alpha\) \(\delta = \beta\)

Step 1: Identify the roots of the given equations

Given the equation \(ax^2 + bx + c = 0\) has roots \(\alpha\) and \(\beta\). Also, we need to find the roots of the equation \(a^3x^2 + abc x + c^3 = 0\), let's denote these roots as \(\gamma\) and \(\delta\).

Step 2: Utilize Vieta's formulas

For the equation \(ax^2 + bx + c = 0\), we have the following relationships: Sum of roots: \( \alpha + \beta = -\frac{b}{a} \) (1) Product of roots: \( \alpha \beta = \frac{c}{a} \) (2) For the equation \(a^3x^2 + abc x + c^3 = 0\), we have the following relationships: Sum of roots: \( \gamma + \delta = -\frac{abc}{a^3} = -\frac{b}{a} \) (3) Product of roots: \( \gamma \delta = \frac{c^3}{a^3} = \frac{c}{a} \) (4)

Step 3: Relate the roots of the two equations

From (1) and (3), we get: \( \alpha + \beta = \gamma + \delta \) From (2) and (4), we get: \( \alpha \beta = \gamma \delta \) Comparing these results, we can conclude the following relationship for the roots of the two quadratic equations: The roots of the equation \(a^3x^2 + abc x + c^3=0\) can be expressed in terms of the roots of the equation \(ax^2 + bx + c=0\) as: \(\gamma = \alpha\) \(\delta = \beta\)

Key Concepts

Vieta's Formulas

Vieta's formulas provide a beautiful connection between the coefficients of a polynomial equation and its roots. Specifically, for a quadratic equation of the form \( ax^2 + bx + c = 0 \), Vieta's formulas state:

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

These relationships arise because of the way quadratic equations factor into expressions involving their roots. By rearranging the terms of the quadratic equation to \( a(x-\alpha)(x-\beta) = 0 \), the coefficients directly reflect these relationships. These simple yet powerful formulas allow us to solve and relate more complex quadratic scenarios easily.

Roots of Quadratic Equations

The roots of quadratic equations hold essential clues about its graph, and they are the points where the quadratic crosses the x-axis. Consider the general form of a quadratic equation: \( ax^2 + bx + c = 0 \). Solving for the roots involves finding distinct points \( x \) such that the equation equals zero. This is often done using the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula gives us two solutions, \( \alpha \) and \( \beta \), known as the roots. Understanding these roots is crucial as they inform us of:

• Whether the parabola intersects the x-axis.
• The nature of the roots: real and distinct, real and repeated, or complex, depending on the discriminant \( b^2 - 4ac \).

Polynomial Relationships

Polynomial relationships are essential for understanding the connections between different polynomial equations and their roots. In this exercise, you are asked to find roots of the polynomial \( a^3x^2 + abc x + c^3 = 0 \) in terms of roots \( \alpha \) and \( \beta \) of another equation. By applying Vieta’s formulas, we derive connections between coefficients and roots.Understanding that polynomial equations of the same degree can share the same roots if their coefficients relate in a particular way is foundational:

• The sum of the roots \( \gamma + \delta \) matches with \( \alpha + \beta \) based on the coefficients.
• The product of the roots \( \gamma \delta \) also reflects the coefficients’ relationship just like \( \alpha \beta \).

These insights allow us to see how different quadratic equations can be interconnected through inherent properties of polynomials.