Deriving the Quadratic Equation with Squared Roots

Understanding how to manipulate quadratic equations to find a new equation with squared roots is a fundamental skill in advanced algebra. This article delves into the process of transforming a given quadratic equation such that its roots are the squares of the original equation's roots. We will explore various steps, including algebraic manipulation, and provide a thorough explanation of each step.

Introduction to the Problem

Consider a quadratic equation of the form ({x}^{2} ax b 0). This equation can be written in the standard form ({ax}^{2} bx c 0) with leading coefficient 1, simplifying our calculations. Let’s denote the roots of this equation as (r_1) and (r_2). According to Vieta's formulas, the sum and product of the roots can be expressed as:

Sum of the roots: (r_1 r_2 -a) Product of the roots: (r_1 r_2 b)

Objective: Derive the New Quadratic Equation

The objective is to derive a new quadratic equation whose roots are the squares of the roots of the original equation. That is, the new roots will be (r_1^2) and (r_2^2).

Step-by-step Derivation

1. **Sum of the new roots:**

Since the roots of the new equation are (r_1^2) and (r_2^2), the sum of the new roots can be found as:

[ r_1^2 r_2^2 (r_1 r_2)^2 - 2r_1r_2 (-a)^2 - 2b a^2 - 2b ]

2. **Product of the new roots:**

The product of the new roots is the square of the product of the original roots:

[ r_1^2 r_2^2 (r_1 r_2)^2 b^2 ]

3. **Constructing the new quadratic equation:**

Using the sum and product of the new roots, the new quadratic equation can be constructed as:

[ x^2 - (a^2 - 2b)x b^2 0 ]

Alternative Derivation Methods

There are alternative methods to derive the same equation, including using polynomial transformation and substitution techniques:

Method 1: Polynomials and Substitution

Let (f(x) x^2 ax b). Considering (y x^2), we can substitute (x^2 y) into the original equation: (y - r_1^2 -a cdot y - b) and (y - r_2^2 -a cdot y - b).

The resulting polynomial (y^2 - 2ay b^2 0) can be simplified to the form:

[ y^2 - 2ay b^2 0 quad text{or} quad y^2 - a^2 - 2b b^2 0 ]

Method 2: Expanding and Identifying Terms

Given (x^2 ax b 0), let (y x^2). Expand and simplify the polynomial (y^2 - 2ay - b^2 a^2y 0).

After rearranging the terms, we obtain:

[ y^2 - a^2 - 2b b^2 0 quad text{or} quad y^2 - 2by - a^2y b^2 0 ]

Conclusion

The problem of finding a quadratic equation whose roots are the squares of the original roots can be systematically solved using algebraic techniques such as polynomial manipulation and substitution. The equation we derived, (x^2 - 2b a^2x b^2 0), encapsulates the logic and the underlying mathematics involved in this transformation. Understanding this process is crucial for tackling more complex problems in algebra and higher mathematics.