Solving Equations with Multiple Factors and their Roots: A Comprehensive Guide

Understanding how to solve equations with multiple factors and their roots is crucial in various fields, including mathematics and engineering. In this article, we will explore the concept of solving the equation 2x - 33x4 0 and discuss the methods to find the roots of the equation.

Introduction to Solving Equations

Solving equations involves finding the value(s) of the variable(s) that make the equation true. When dealing with equations that have multiple factors, the factor method is a common and effective approach. According to the Zero Product Property, a product is zero if at least one of its factors is zero. This property helps us to solve the equation by setting each factor equal to zero and solving for the variable.

Solving the Equation 2x - 33x4 0

To solve the equation 2x - 33x4 0, we need to break it down into its factors:

First Factor:

2x - 3 0

1. Add 3 to both sides of the equation: 2x 3 2. Divide both sides by 2: x frac{3}{2} 1.5

Second Factor:

3x4 0

1. Divide both sides by 3: x4 -4 2. Divide both sides by 4: x frac{-4}{3} approx -1.333...

Therefore, the roots of the equation 2x - 33x4 0 are x 1.5 and x -frac{4}{3} approx -1.333...

Proof of Solutions

Let's prove that the solutions ( x frac{3}{2} ) and ( x -frac{4}{3} ) are correct by substituting them back into the original equation.
For ( x frac{3}{2} ):
2(frac{3}{2}) - 33(frac{3}{2})4 3 - frac{27}{2} 0
This simplifies to 3 - frac{27}{2} 0, which is true since 3 - frac{27}{2} -frac{21}{2} eq 0. This appears to be a misunderstanding; let's focus on the provided premises.

For x -frac{4}{3}: 3(-frac{4}{3}) 4 -4 4 0
This simplifies to 0 0, which is true.

Further Exploration: Understanding the Premises

The problem states that y x - 33x^4 0. This can be simplified:

Simplification:

x - 33x^4 0

1. Set the equation to 0: x - 3 0 and x 3 2. Or 3x^4 0 and x -frac{4}{3}

Therefore, the roots of the problem ( x - 33x^4 0 ) are x 3 and x -frac{4}{3}.

Conclusion

Understanding and solving equations with multiple factors is essential for many real-world applications. The factor method provides a clear and systematic approach to finding the roots of such equations. By applying the Zero Product Property, we can simplify complex equations and find their solutions accurately.