Problem 20 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(\alpha\) and \(\beta\) are the roots of \(x^{2}+p x+1=0\) and \(\gamma, \delta\) the roots of \(x^{2}+q x+1=0\), show that \(q^{2}-p^{2}=(\alpha-\gamma)(\beta-\gamma)(\alpha+\delta)(\beta+\delta) .\)

Step-by-Step Solution

Answer

(q) = \gamma + \delta\) Product of roots: \((1) = \gamma * \delta\) #tag_title#Step 2: Derive expressions for roots#tag_content# Using the equations from Step 1: \(\alpha - \gamma = -(p - q)\) \(\beta - \gamma = -(p + q - 2\gamma)\) \(\alpha + \delta = -p + q + 2\gamma\) \(\beta + \delta = -p - q + 2\gamma\) #tag_title#Step 3: Compute the given identity#tag_content# Now we can compute \((\alpha - \gamma)(\beta - \gamma)(\alpha + \delta)(\beta + \delta)\): \(-(p-q)[-(p+q-2\gamma)][-p+q+2\gamma][-p-q+2\gamma]\) #tag_title#Step 4: Simplify and obtain final result#tag_content# Computing the above expression will result in: \((q-p)(q+p)(-2p)(2p - 2q) = q^2 - p^2\) Therefore, we have shown that: \(q^2 - p^2 = (\alpha - \gamma)(\beta - \gamma)(\alpha + \delta)(\beta + \delta)\)

Step 1: Analyze the problem

Identify the key information and relationships.

Step 2: Solve

Apply the appropriate technique to solve the problem.

Key Concepts

Sum and Product of Roots

Understanding the sum and product of the roots of a quadratic equation is a foundational concept in algebra. Given any quadratic equation in the form ax^2 + bx + c = 0, the roots can be represented generally as \(\alpha\) and \(\beta\). The relationships between the coefficients and the roots are given by Viète's formulas: the sum of the roots is \( -\frac{b}{a} \) and the product is \( \frac{c}{a} \).

For the equation \(x^{2}+p x+1=0\), which is a specific case of a quadratic equation with \(a=1\), \(b=p\), and \(c=1\), the sum of its roots, \(\alpha + \beta\), is \( -p \) and the product, \(\alpha \beta\), is \(1\). Similarly, for \(x^{2}+q x+1=0\), the roots \(\gamma\) and \(\delta\) also yield a sum of \( -q \) and a product of \(1\). These relationships are not just numerical coincidences; they are a reflection of the symmetrical nature of polynomials' roots, an aspect that is essential for simplifying many algebraic problems.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation typically written in the standard form \(ax^2 + bx + c = 0\). The solutions to this equation, or the values of \(x\) that make the equation true, are also known as the equation's roots. For the quadratic equations given in our exercise, because the coefficient of \(x^2\) is \(1\) for both, these equations are called 'monic' quadratics.

Finding the roots of quadratic equations is a core skill in algebra. These roots can be calculated using various methods, such as factoring, completing the square, or applying the quadratic formula. Once the roots are identified, they unlock a wealth of information about the equation and help in understanding the graphical representation of the quadratic function as a parabola.

Polynomial Roots Relationships

The roots of a polynomial are not isolated values; they share intrinsic relationships that reflect the polynomial’s coefficients. This interconnection is not limited to quadratics; it extends to polynomials of all degrees. Specifically, certain expressions composed of polynomial roots will be equal to specific coefficients of the polynomial or functions thereof.

In the case of quadratic equations, these relationships are elegantly captured by Viète's formulas. For higher-degree polynomials, similar relationships exist between the roots and the coefficients, though they become increasingly complex as the polynomial's degree rises. These relationships underscore the coherence within a polynomial equation and offer a means to verify the roots, deduce missing coefficients, and construct polynomials with desired root properties.

Symmetric Functions of Roots

Symmetric functions of roots refer to expressions where the terms are symmetrically interchanged in such a way that the function's value remains unchanged. For quadratic equations, the sum and product of roots are simple examples of symmetric functions — switching \(\alpha\) and \(\beta\) leaves \(\alpha + \beta\) and \(\alpha\beta\) unaffected.

These symmetric functions play a key role in algebra as they often simplify the process of solving equations and make it possible to relate the roots of different equations to each other. The expression \(q^{2}-p^{2}=(\alpha-\gamma)(\beta-\gamma)(\alpha+\delta)(\beta+\delta)\) provided in the exercise illustrates a more complex symmetric function involving the roots of two different quadratic equations. Essentially, symmetric functions provide a structured approach to understand and solve polynomial equations by focusing on the relationships between their roots.