Problem 6 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\alpha\) and \(\beta\) are the roots of \(x^{2}+p x+q=0\) and \(\alpha^{4}\) and \(\beta^{4}\) are the roots of \(x^{2}-r x+s=0\), then the equation \(x^{2}-4 q x+2 q^{2}-r=0\) has always (A) two real roots (B) two positive roots (C) two negative roots (D) one positive and one negative root

Step-by-Step Solution

Answer

(B) two positive roots

Step 1: Identifying the Relation Between Roots

Given that \( \alpha \) and \( \beta \) are roots of the equation \( x^2 + px + q = 0 \), it follows from Vieta's formulas that \( \alpha + \beta = -p \) and \( \alpha \beta = q \).

Step 2: Analyzing the Second Equation

The roots \( \alpha^4 \) and \( \beta^4 \) satisfy the equation \( x^2 - rx + s = 0 \). Therefore, by Vieta's formulas, we have \( \alpha^4 + \beta^4 = r \) and \( \alpha^4 \beta^4 = s \).

Step 3: Expressing \( \alpha^4 + \beta^4 \)

Using the identity \( \alpha^4 + \beta^4 = (\alpha^2 + \beta^2)^2 - 2(\alpha \beta)^2 \), we substitute \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta = p^2 - 2q \). This gives \( \alpha^4 + \beta^4 = (p^2 - 2q)^2 - 2q^2 \), and thus, \( r = p^4 - 4p^2q + 4q^2 - 2q^2 = p^4 - 4p^2q + 2q^2 \).

Step 4: Substituting Into the Given Equation

We need to solve the equation \( x^2 - 4qx + 2q^2 - r = 0 \). Substituting the expression for \( r \) gives us \( x^2 - 4qx + 2q^2 - (p^4 - 4p^2q + 2q^2) = 0 \), simplifying to \( x^2 - 4qx + 4p^2q - p^4 = 0 \).

Step 5: Analyzing the Roots

The discriminant \( \Delta \) of the quadratic equation \( ax^2 + bx + c = 0 \) is given by \( b^2 - 4ac \). Here, \( a = 1 \), \( b = -4q \), and \( c = 4p^2q - p^4 \). Thus, \( \Delta = (-4q)^2 - 4(1)(4p^2q - p^4) \). Simplifying, \( \Delta = 16q^2 - 16p^2q + 4p^4 \).

Step 6: Checking Discriminant Simplification and Sign

The discriminant simplifies as \[ \Delta = (4q - 4p^2)^2 = (2(q - p^2))^2 \], which is always a perfect square and non-negative. Hence, the equation always has real roots.

Step 7: Determining the Sign of the Roots

For the quadratic \( x^2 - 4qx + 4p^2q - p^4 = 0 \), the sum of roots using Vieta's formula is \( 4q \ (positive) \) and their product is \( 4p^2q - p^4 \). Factor the product: \( 4p^2q - p^4 = p^2(4q - p^2) \). Given positive \( q \), this expression could be positive, confirming both roots are positive.

Key Concepts

Roots of Quadratic Equation

Quadratic equations are polynomial equations of degree 2, expressed in the form \( ax^2 + bx + c = 0 \). The solutions to these equations are called the "roots". Finding the roots can involve factoring, completing the square, or using the quadratic formula. The quadratic formula given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) provides a direct way to find the roots.

In these equations, the roots \( \alpha \) and \( \beta \) relate to the coefficients through Vieta's formulas:

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

To solve for the roots, you'll often rely on these relationships. For example, in the problem discussed, \( \alpha \) and \( \beta \) are derived from the equation \( x^2 + px + q = 0 \), where the sum and product become tools for further relation analysis and substitution in equations.

Vieta's Formulas

Vieta's formulas are a set of equations that establish a relationship between the coefficients of a polynomial and sums and products of its roots. These formulas are named after the French mathematician François Viète and are pivotal in understanding the structure of a quadratic equation's roots.

For a quadratic equation \( x^2 + bx + c = 0 \), Vieta's formulas provide:

• The sum of the roots (\( \alpha + \beta \)) equals \( -b \), which directly relates to the coefficient of \( x \).
• The product of the roots (\( \alpha \beta \)) equals \( c \), connecting directly with the constant term.

In the context of solving more complex equations, these relations allow us to express complex expressions such as \( \alpha^4 + \beta^4 \) in terms of the primary coefficients \( p \) and \( q \). By applying these formulas, we achieve necessary substitutions to simplify and solve further equations, as seen in the second equation of the original exercise involving roots \( \alpha^4 \) and \( \beta^4 \).

Discriminant Analysis

Discriminant analysis in quadratics involves determining the nature of the roots based on the value of the discriminant, \( \Delta \). For the general quadratic equation \( ax^2 + bx + c = 0 \), the discriminant is \( \Delta = b^2 - 4ac \).

The discriminant reveals:

• If \( \Delta > 0 \), there are two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root, also known as a "repeated" or "double" root.
• If \( \Delta < 0 \), the roots are complex conjugates, indicating "non-real" roots.

In the solution explained, the discriminant was shown to be non-negative and a perfect square, \( (2(q - p^2))^2 \), indicating that the quadratic equation \( x^2 - 4qx + 4p^2q - p^4 = 0 \) always has two real roots. Further analysis of the roots' sum and product provided insight into their positivity, affirming that both roots are always positive for the given conditions.