Solving Complex Algebraic Equations: A Step-by-Step Guide

Introduction

When dealing with complex algebraic equations, the process can often feel daunting. However, by breaking down the equation into simpler parts and systematically solving each component, you can tackle the problem effectively. This guide will walk you through solving the equation (9x^2 - 19x^2 - 1) (3x - 1 - 3x) step by step using techniques such as simplification, factorization, and equation solving.

Solving the Equation: Step-by-Step

Given the equation: [(9x^2 - 19x^2 - 1) (3x - 1 - 3x)]

Step 1: Simplify Both Sides

First, let's simplify the left side of the equation. We can use the difference of squares to simplify the expression:

[9x^2 - 19x^2 - 1 (3x - 1)(3x 1) - 1]

Applying the difference of squares formula:

[9x^2 - 1 (3x - 1)(3x 1) - (3x - 1)]

Now, let's simplify the right side of the equation:

[3x - 1 - 3x -1]

Thus, the equation becomes:

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

Step 2: Factorize the Equation

Notice that (3 - x) is a common factor:

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

Simplify the expression inside the parentheses:

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

Step 3: Solve for x

Divide both sides by (3x) (assuming (3x eq 0)):

[(3x - 1) -1/3x]

This leads to:

[3x - 1 -1/3x]

Multiplying both sides by 3x to clear the fraction:

[9x^2 - 3x -1]

Adding 1 to both sides:

[9x^2 - 3x 1 0]

Now, we can solve for (x) using the quadratic formula (ax^2 bx c 0), where (a 9), (b -3), and (c 1):

[x frac{-b pm sqrt{b^2 - 4ac}}{2a}]

Substituting the values:

[x frac{3 pm sqrt{(-3)^2 - 4(9)(1)}}{2(9)}]

[x frac{3 pm sqrt{9 - 36}}{18}]

[x frac{3 pm sqrt{-27}}{18}]

Since the discriminant is negative, there are no real solutions for (x).

Conclusion:

Given the complexity and the nature of the equation, we conclude that there are no real solutions.

Keywords and Related Terms

This guide has covered techniques for solving complex algebraic equations, including simplification, factoring, and solving quadratic equations. Key terms include:

algebraic equations: equations involving variables and constants. solving equations: methods for isolating and finding the values of variables that satisfy the equation. mathematical problems: problems that involve the use of mathematical operations to find a solution.

Conclusion

While the given equation did not yield a real solution, understanding the steps and techniques used in the process is invaluable for solving more complex algebraic equations in the future. Stay tuned for more detailed guides and extensive resources to enhance your problem-solving skills in algebra.