Solving Complex Algebraic Equations Involving Radical Expressions

Algebra often presents challenges in solving complex equations, especially those involving radical expressions. This article will walk through a detailed example of how to solve an equation of the form (x^2 frac{1}{x^2} a - bsqrt{c}), using rationalization and other algebraic techniques.

Introduction

The given equation involves a term with a radical expression in the denominator, which can be simplified using rationalization. Rationalization is a process that eliminates radicals from the denominator, making the equation more manageable. In this article, we will explore the steps necessary to solve the equation (x^2 frac{1}{x^2} 5 - 2sqrt{6}).

Rationalization and Simplification

First, we start with the given equation:

(x^2 frac{1}{x^2} 5 - 2sqrt{6})

To simplify the right-hand side, we use the formula for the difference of squares:

((a - b)^2 a^2 - 2ab b^2)

In this case, (a 5) and (b sqrt{6}), so:

((5 - 2sqrt{6})^2 25 - 2 cdot 5 cdot 2sqrt{6} (2sqrt{6})^2)

This simplifies to:

(25 - 20sqrt{6} 24 49 - 20sqrt{6})

Hence, the equation becomes:

(x^2 frac{1}{x^2} 5 - 2sqrt{6})

We rationalize the equation by multiplying the numerator and the denominator by the conjugate of the denominator:

(x^2 frac{1}{x^2} 5 - 2sqrt{6})

Now, multiply both the numerator and denominator by (5 2sqrt{6}):

(x^2 frac{1}{x^2} (5 - 2sqrt{6})(5 2sqrt{6}) 25 - 24 1)

This simplifies to:

(x^2 frac{1}{x^2} 10)

Step-by-Step Solution

Step 1: Simplify the given expression.

(x^2 5 - 2sqrt{6})

Step 2: Find the conjugate of the denominator.

The conjugate of (5 - 2sqrt{6}) is (5 2sqrt{6}).

Step 3: Rationalize the expression.

(frac{1}{5-2sqrt{6}} frac{5 2sqrt{6}}{1})

Step 4: Combine the expressions.

((5 - 2sqrt{6}) (5 2sqrt{6}) 10)

Conclusion

To solve complex algebraic equations involving radical expressions, rationalization is a powerful technique. By breaking down the problem into simpler steps and using algebraic identities, the solution can be found efficiently. This method is particularly useful for solving equations with radical terms in the denominator.

For more detailed examples and step-by-step solutions, refer to the following resources:

Detailed Algebraic Equations Solver Examples of Rationalization in Mathematics Advanced Algebra Tutorial

Understanding and applying these techniques will greatly enhance your problem-solving skills in algebra and beyond.