How to Solve the Equation and Find Its Roots

In this article, we will explore the detailed process of finding the roots of a complex algebraic equation. We will use the given equation and methodically break down the steps to uncover its solutions, emphasizing the importance of checking for extraneous solutions. This will be particularly useful for individuals studying advanced algebra or preparing for competitions where understanding such techniques is crucial.

Introduction to the Problem

Given the equation:

[ xsqrt{x^2sqrt{x^3-1}} 1 ]

We start by setting up the equation step-by-step, deriving solutions, and then validating them to ensure accuracy.

Step-by-Step Solution

The initial step is to isolate the equation in a manageable form. Let's start by rearranging the equation and then proceed with a series of algebraic manipulations:

Given: [ xsqrt{x^2sqrt{x^3-1}} 1 ] Subtracting ( x ) from both sides gives:

[ xsqrt{x^2sqrt{x^3-1}} - x 1 - x ]

By squaring both sides, we can eliminate the first square root:

[ x^2sqrt{x^3-1} (1 - x)^2 ]

Next, squaring again to remove the second square root:

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

This leads to the polynomial equation:

[ x^3 - 4x^2 4x 0 ]

Factoring the polynomial equation:

[ x(x^2 - 4x 4) 0 ]

Which simplifies to:

[ x(x - 2)^2 0 ]

Thus, the solutions are:

[ x 0 quad text{or} quad x 2 ]

Checking for Extraneous Solutions

When solving algebraic equations by squaring or other operations, it's essential to verify the solutions to ensure they are valid. Here, we check the potential solutions to see if they satisfy the original equation:

x 0

[ 0sqrt{0^2sqrt{0^3-1}} 1 ] [ 0 1 ]

This is not true, so ( x 0 ) is not a valid solution in the original context.

x 2

[ 2sqrt{4sqrt{81}} 1 ] [ 2sqrt{4cdot 9} 1 ] [ 2sqrt{36} 1 ] [ 2 cdot 6 1 ] [ 12 1 ]

This is also not true, indicating that ( x 2 ) is an extraneous solution.

Concluding Remark

The only valid solution, verified through the checking process, is ( x 0 ). This demonstrates the importance of validating solutions when solving complex algebraic equations, especially when solving through squaring operations.

In summary, the equation [ xsqrt{x^2sqrt{x^3-1}} 1 ] has a sole valid root at ( x 0 ). This example highlights the systematic approach to solving such equations and the necessity to check for extraneous solutions.

Understanding these techniques is beneficial for students and professionals in various fields such as engineering, physics, and mathematics.