Problem 4 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If \(\alpha\) and \(\beta\) be the roots of the equation \(p x^{2}\) \(+q x+r=0 .\) Hence, obtain the equation whose roots are \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\).

Step-by-Step Solution

Answer

The equation is \( x^2 - \frac{q^2 - 2pr}{r} x + 1 = 0 \).

Step 1: Understand the Vieta's formulas

For a quadratic equation of the form \( p x^2 + q x + r = 0 \), Vieta's formulas state that the sum of the roots \( \alpha + \beta = -\frac{q}{p} \) and the product of the roots \( \alpha \beta = \frac{r}{p} \). This will be useful in finding the new equation.

Step 2: Find the Sum and Product of New Roots

The new roots are \( \frac{\alpha}{\beta} \) and \( \frac{\beta}{\alpha} \). To find the sum of these roots, use: \( \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha\beta} \), and for the product, use: \( \frac{\alpha}{\beta} \cdot \frac{\beta}{\alpha} = 1 \).

Step 3: Express \( \alpha^2 + \beta^2 \) in Terms of Known Quantities

Recall the identity \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \). Plug in the values from Vieta's formulas to get: \( \alpha^2 + \beta^2 = \left(-\frac{q}{p}\right)^2 - 2\left(\frac{r}{p}\right) = \frac{q^2}{p^2} - \frac{2r}{p} \).

Step 4: Form the New Quadratic Equation

Using the sum and product of the new roots, the new equation is \( x^2 - Sx + P = 0 \), where \( S = \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha\beta} = \frac{\frac{q^2}{p^2} - \frac{2r}{p}}{\frac{r}{p}} \) and \( P = 1 \).

Step 5: Simplify the Coefficients

The simplified expression for \( S \) becomes \( \frac{\frac{q^2 - 2pr}{pr}}{\frac{r}{p}} = \frac{q^2 - 2pr}{pr} \times \frac{p}{r} = \frac{q^2 - 2pr}{r} \). Hence, the equation is \( x^2 - \frac{q^2 - 2pr}{r} x + 1 = 0 \).

Key Concepts

Quadratic Equations

Quadratic equations form a central part of algebra, and they often come up in various mathematical contexts. In general, a quadratic equation is expressed in the form:

• \( px^2 + qx + r = 0 \)

where \( p \), \( q \), and \( r \) are constants, and \( x \) represents the variable or unknown we are solving for. Quadratic equations typically have two roots, which are the values of \( x \) that satisfy the equation. To solve these equations, we often use the quadratic formula, derived by completing the square, given by:

• \( x = \frac{-q \pm \sqrt{q^2 - 4pr}}{2p} \).

The solutions or roots \( \alpha \) and \( \beta \) can be real or complex depending on the discriminant, \( q^2 - 4pr \). If you have a firm understanding of quadratic equations, you can work backward from the roots to the original equation using Vieta's formulas. These formulas show the relationships between the roots and the coefficients of the equation.

Roots of Polynomial

The roots of a polynomial are crucial to understanding its behavior. For a quadratic polynomial, like our example \( px^2 + qx + r = 0 \), the roots \( \alpha \) and \( \beta \) are the values where the polynomial equals zero. These roots are not just solutions to the equation but also represent points where the graph of the polynomial (typically a parabola for quadratics) intersects the x-axis.

Vieta’s formulas relate the coefficients of the polynomial to the sum and product of its roots. Specifically, for a quadratic:

• The sum of the roots is given by \( \alpha + \beta = -\frac{q}{p} \), and
• The product of the roots is \( \alpha \beta = \frac{r}{p} \).

Using these relationships helps in forming new equations when the roots are transformed. In our exercise, when we derive the new roots as \( \frac{\alpha}{\beta} \) and \( \frac{\beta}{\alpha} \), knowing these original relationships aids us in calculating the new sum and product, leading to the correct formation of the derived quadratic equation.

Algebraic Identities

Algebraic identities are equations that hold for all possible values of the variables within them. In our context, they are vital tools for simplifying and transforming expressions. One key identity used in the solution of this exercise is:

• \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \).

This specific identity allowed us to express \( \alpha^2 + \beta^2 \) in terms of the sum and product of the roots, which were initially obtained from Vieta's formulas.

By expanding and rearranging these identities, we can derive new expressions that help us form equations with warped root characters. Another concept touched on was simplifying algebraic fractions, where rewriting complex terms can reveal underlying simpler forms, such as transforming \( \frac{\frac{q^2 - 2pr}{pr}}{\frac{r}{p}} \) into \( \frac{q^2 - 2pr}{r} \). Understanding algebraic identities and their applications truly enhances one's ability to handle polynomial transformations and equation derivations efficiently.