What is the LCM of the Expression 2x2-5x3?

In this article, we will explore the process of factoring the algebraic expression 2x2-5x3 to find its roots. We will also delve into the concept of the Least Common Multiple (LCM) in the context of algebraic expressions. This article is designed to help students and educators understand the techniques involved in solving quadratic equations through factoring and the importance of LCM in related algebraic manipulations.

Understanding the Expression 2x2-5x3

Let's start with the given expression: 2x2-5x3. To factor this expression, it's essential to recognize the structure: ax2 bx c. In this case, a 2, b -5, and c -3.

Step 1: Multiply the Outside Factors a and c

To find the LCM (or, in this context, the coefficients that can help us factor), we start by multiplying the outside factors a and c:

2 × -3 -6

Next, we need two numbers that multiply to give -6 and add to give the coefficient of the middle term, -5.

Finding the Correct Numbers

We need to find two numbers that multiply to -6 and add to -5. Let's test the possible pairs:

-2 × 3 -6, but -2 3 1 -3 × 2 -6, and -3 2 -1 -6 × 1 -6, and -6 1 -5

The correct pair is -6 and 1, which add to -5 and multiply to -6.

Step 2: Split the Middle Term Using the Found Numbers

Now, we split the middle term using the numbers -6 and 1:

2x2 -6x 1x - 3

Grouping the terms, we get:

(2x2 -6x) (1x - 3)

Factoring by Grouping

Next, we factor out the greatest common factor from each group:

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

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

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

This is the factored form of the expression 2x2-5x3.

Finding the Roots of the Expression

Setting each factor to zero provides the roots of the equation:

2x 1 0 x - 3 0

Solving these equations:

2x 1 0 2x -1 x -1/2 x - 3 0 x 3

The Importance of the LCM in Algebraic Equations

The Least Common Multiple (LCM) is crucial in understanding the roots and factoring of algebraic expressions. By finding the LCM, we can simplify complex equations and solve for variables more efficiently. In the context of quadratic equations, the LCM helps us identify factors that can be used to break down the expression and find its roots.

Conclusion

In conclusion, the expression 2x2-5x3 can be factored as (2x 1)(x - 3). The process involves multiplying the outside factors, finding the correct pair of numbers, and then using grouping and factoring techniques to simplify the expression. Understanding the LCM is essential for manipulating and solving quadratic equations.

Further Reading

For more information on LCM, algebraic expressions, and factoring techniques, consider exploring the following resources:

Algebra Textbooks Online Math Courses on platforms like Khan Academy Math Forums and Discussion Boards

Key Takeaways

The LCM is a fundamental concept in algebra. Factoring techniques are essential for solving quadratic equations. Understanding the structure of expressions like ax2 bx c is crucial for simplifying and solving algebraic equations.