Solving the Equation (x^2 - 1^2 x - 1^2): A Detailed Guide

Introduction

Quadratic equations are a fundamental concept in algebra, and solving them efficiently is essential for many practical applications. This article provides a detailed explanation of how to solve a specific quadratic equation x^2 - 1^2 x - 1^2.

Step-by-Step Solution

We start with the equation:

x^2 - 1^2 x - 1^2

Expand Both Sides of the Equation

The first step is to expand both sides of the equation:

Left-Hand Side: x^2 - 1^2 x^2 - 1 Right-Hand Side: x - 1^2 x - 1

Equating the two sides, we get:

x^2 - 1 x - 1

Subtract x - 1 from Both Sides

By subtracting x - 1 from both sides, we simplify the equation:

x^2 - 1 - (x - 1) 0

Simplifying the left side gives:

x^2 - x 0

Factor the Quadratic Equation

To solve this quadratic equation, we factor it:

x(x - 1) 0

This gives us the following solutions:

x 0 x - 1 0 rarr; x 1

Verify Solutions

Let's verify these solutions by plugging them back into the original equation:

For x 0: 0^2 - 1 0 - 1 -1 -1 For x 1: 1^2 - 1 1 - 1 0 0

Advanced Techniques for Solving Quadratic Equations

For more complex quadratic equations, advanced techniques like the Rational Root Theorem can be useful. Let's apply these techniques to the given equation x^4 - 3x^2 - 2x 0 after expanding it:

Rational Root Theorem

The Rational Root Theorem suggests that any rational solution of the polynomial equation must be a factor of the constant term divided by a factor of the leading coefficient. In this equation, the constant term is -2 and the leading coefficient is 1. Therefore, the potential rational roots are:

1 -1 2 -2

Testing for Rational Roots

Let's test x 1:

1^3 - 3(1) - 2 1 - 3 - 2 -4 ne; 0

This shows that x 1 is not a root. However, let's test x -1:

(-1)^3 - 3(-1) - 2 -1 3 - 2 0

Thus, x -1 is a root of the equation.

Factor the Cubic Polynomial

Using synthetic division or polynomial long division, we divide x^3 - 3x - 2 by x 1:

x^3 - 3x - 2 (x 1)(x^2 - x - 2)

Factor the Quadratic Polynomial

The quadratic polynomial x^2 - x - 2 can be factored as:

x^2 - x - 2 (x 1)(x - 2)

Combine All Factors

Using the factors from the previous steps, the complete factorization is:

x(x 1)^2(x - 2) 0