Simplify The Expression: X 2 + 3 X X − 2 ⋅ X 2 − 5 X + 6 4 X \frac{x^2 + 3x}{x - 2} \cdot \frac{x^2 - 5x + 6}{4x} X − 2 X 2 + 3 X ⋅ 4 X X 2 − 5 X + 6

Mar 11, 2025 by ADMIN 154 views

**Simplify the Expression: A Step-by-Step Guide**

Introduction

In this article, we will simplify the given expression: $\frac{x^2 + 3x}{x - 2} \cdot \frac{x^2 - 5x + 6}{4x}$ . This involves factoring the numerator and denominator, canceling out common factors, and simplifying the resulting expression. We will break down the process into manageable steps, making it easier to understand and follow along.

Step 1: Factor the Numerator and Denominator

To simplify the expression, we need to factor the numerator and denominator. The numerator is $x^2 + 3x$ , and the denominator is $x - 2$ . We can factor the numerator as follows:

$x^2 + 3x = x(x + 3)$

The denominator is already factored as $x - 2$ .

The second fraction has a numerator of $x^2 - 5x + 6$ and a denominator of $4x$ . We can factor the numerator as follows:

$x^2 - 5x + 6 = (x - 2)(x - 3)$

The denominator is already factored as $4x$ .

Step 2: Rewrite the Expression with Factored Numerator and Denominator

Now that we have factored the numerator and denominator, we can rewrite the expression as follows:

$\frac{x(x + 3)}{x - 2} \cdot \frac{(x - 2)(x - 3)}{4x}$

Step 3: Cancel Out Common Factors

We can see that the expression $(x - 2)$ appears in both the numerator and denominator. We can cancel out this common factor:

$\frac{x(x + 3)}{1} \cdot \frac{(x - 3)}{4x}$

Step 4: Simplify the Expression

Now that we have canceled out the common factor, we can simplify the expression by multiplying the numerators and denominators:

$\frac{x(x + 3)(x - 3)}{4x}$

Step 5: Cancel Out Common Factors Again

We can see that the expression $x$ appears in both the numerator and denominator. We can cancel out this common factor:

$\frac{(x + 3)(x - 3)}{4}$

Conclusion

In this article, we simplified the given expression: $\frac{x^2 + 3x}{x - 2} \cdot \frac{x^2 - 5x + 6}{4x}$ . We factored the numerator and denominator, canceled out common factors, and simplified the resulting expression. The final simplified expression is $\frac{(x + 3)(x - 3)}{4}$ .

Tips and Tricks

When simplifying expressions, it's essential to factor the numerator and denominator to cancel out common factors.
Make sure to cancel out common factors carefully to avoid errors.
Simplifying expressions can be a complex process, but breaking it down into manageable steps can make it easier to understand and follow along.

Real-World Applications

Simplifying expressions is a crucial skill in mathematics, and it has many real-world applications. For example, in physics, simplifying expressions can help us understand complex phenomena, such as the motion of objects. In engineering, simplifying expressions can help us design and optimize systems, such as electrical circuits.

Common Mistakes to Avoid

When simplifying expressions, it's essential to avoid common mistakes, such as:

Not factoring the numerator and denominator
Not canceling out common factors carefully
Not simplifying the expression correctly

By following the steps outlined in this article and avoiding common mistakes, you can simplify expressions with confidence and accuracy.

Final Thoughts

Introduction

In our previous article, we simplified the given expression: $\frac{x^2 + 3x}{x - 2} \cdot \frac{x^2 - 5x + 6}{4x}$ . We factored the numerator and denominator, canceled out common factors, and simplified the resulting expression. In this article, we will answer some frequently asked questions about simplifying expressions.

Q: What is the first step in simplifying an expression?

A: The first step in simplifying an expression is to factor the numerator and denominator. This involves breaking down the expression into its simplest form by identifying common factors.

Q: How do I know if I have factored the numerator and denominator correctly?

A: To ensure that you have factored the numerator and denominator correctly, you can use the following steps:

Check if the numerator and denominator can be factored using the distributive property.
Look for common factors, such as greatest common factors (GCFs) or common binomials.
Use the factored form to simplify the expression.

Q: What is the difference between canceling out common factors and simplifying an expression?

A: Canceling out common factors involves removing common factors from the numerator and denominator, while simplifying an expression involves reducing the expression to its simplest form by combining like terms.

Q: Can I simplify an expression by canceling out common factors without factoring the numerator and denominator?

A: No, you cannot simplify an expression by canceling out common factors without factoring the numerator and denominator. Factoring the numerator and denominator is essential to identify common factors and simplify the expression correctly.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

Not factoring the numerator and denominator
Not canceling out common factors carefully
Not simplifying the expression correctly
Not checking for common factors

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by:

Working through examples and exercises
Using online resources and tools
Practicing with different types of expressions, such as rational expressions and polynomial expressions
Joining a study group or working with a tutor

Q: What are some real-world applications of simplifying expressions?

A: Simplifying expressions has many real-world applications, including:

Physics: Simplifying expressions can help us understand complex phenomena, such as the motion of objects.
Engineering: Simplifying expressions can help us design and optimize systems, such as electrical circuits.
Computer Science: Simplifying expressions can help us develop efficient algorithms and data structures.

Q: Can I use technology to simplify expressions?

A: Yes, you can use technology to simplify expressions. Many calculators and computer algebra systems (CAS) can simplify expressions automatically. However, it's essential to understand the underlying mathematics and be able to simplify expressions manually.

Conclusion

Simplifying expressions is a fundamental skill in mathematics, and it has many real-world applications. By following the steps outlined in this article and practicing regularly, you can become proficient in simplifying expressions and apply this skill to a wide range of problems.