Simplify The Expression: 6 X 2 + 9 X + 20 ⋅ 8 X + 40 6 X − 12 \frac{6}{x^2+9x+20} \cdot \frac{8x+40}{6x-12} X 2 + 9 X + 20 6 ⋅ 6 X − 12 8 X + 40

Mar 11, 2025 by ADMIN 149 views

Simplify the Expression: A Step-by-Step Guide to Algebraic Manipulation

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Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities more efficiently. One of the most common techniques used to simplify expressions is factoring. In this article, we will focus on simplifying a given expression using factoring and other algebraic manipulations.

The Given Expression

The given expression is:

$\frac{6}{x^2+9x+20} \cdot \frac{8x+40}{6x-12}$

Our goal is to simplify this expression by factoring the numerator and denominator, and then canceling out any common factors.

Factoring the Numerator and Denominator

Let's start by factoring the numerator and denominator separately.

Factoring the Numerator

The numerator is $8x+40$ . We can factor out the greatest common factor (GCF), which is 8:

$8x+40 = 8(x+5)$

Factoring the Denominator

The denominator is $x^2+9x+20$ . We can factor this quadratic expression by finding two numbers whose product is 20 and whose sum is 9. The numbers are 5 and 4, so we can write:

$x^2+9x+20 = (x+5)(x+4)$

Factoring the Expression

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

$\frac{6}{(x+5)(x+4)} \cdot \frac{8(x+5)}{6(x-2)}$

Canceling Out Common Factors

We can see that the expression contains two common factors: $(x+5)$ and 6. We can cancel out these factors by dividing both the numerator and denominator by $(x+5)$ and 6:

$\frac{6}{(x+5)(x+4)} \cdot \frac{8(x+5)}{6(x-2)} = \frac{8}{(x+4)} \cdot \frac{1}{(x-2)}$

Final Simplification

The expression is now simplified, and we can see that it contains two separate fractions:

$\frac{8}{(x+4)} \cdot \frac{1}{(x-2)} = \frac{8}{(x+4)(x-2)}$

Conclusion

In this article, we simplified the given expression using factoring and other algebraic manipulations. We factored the numerator and denominator, canceled out common factors, and arrived at the final simplified expression. This technique is essential in algebra, as it helps us solve equations and inequalities more efficiently.

Tips and Tricks

When factoring the numerator and denominator, look for the greatest common factor (GCF) and factor it out.
Use the distributive property to expand the expression and simplify it.
Cancel out common factors by dividing both the numerator and denominator by the common factor.
Use the final simplified expression to solve equations and inequalities.

Common Mistakes to Avoid

Failing to factor the numerator and denominator properly.
Not canceling out common factors.
Not using the distributive property to expand the expression.
Not simplifying the expression further.

Real-World Applications

Simplifying expressions is a crucial skill in many real-world applications, including:

Physics: Simplifying expressions is essential in physics to solve equations and inequalities related to motion, energy, and momentum.
Engineering: Simplifying expressions is crucial in engineering to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Simplifying expressions is essential in economics to model and analyze economic systems, such as supply and demand.

Final Thoughts

Simplifying expressions is a fundamental skill in algebra that helps us solve equations and inequalities more efficiently. By factoring the numerator and denominator, canceling out common factors, and using the distributive property, we can simplify expressions and arrive at the final solution. This technique is essential in many real-world applications, including physics, engineering, and economics.

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Introduction

In our previous article, we simplified the expression $\frac{6}{x^2+9x+20} \cdot \frac{8x+40}{6x-12}$ using factoring and other algebraic manipulations. In this article, we will answer some common questions related to simplifying expressions and provide additional tips and tricks to help you master this technique.

Q&A

Q: What is the greatest common factor (GCF) and how do I find it?

A: The greatest common factor (GCF) is the largest factor that divides two or more numbers without leaving a remainder. To find the GCF, look for the largest factor that divides both numbers.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, look for two numbers whose product is the constant term and whose sum is the coefficient of the middle term. Then, write the expression as the product of two binomials.

Q: What is the distributive property and how do I use it?

A: The distributive property is a rule that allows us to expand an expression by multiplying each term inside the parentheses by the term outside the parentheses. To use the distributive property, multiply each term inside the parentheses by the term outside the parentheses.

Q: How do I cancel out common factors?

A: To cancel out common factors, divide both the numerator and denominator by the common factor. This will simplify the expression and eliminate the common factor.

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

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

Failing to factor the numerator and denominator properly
Not canceling out common factors
Not using the distributive property to expand the expression
Not simplifying the expression further

Q: How do I apply simplifying expressions in real-world applications?

A: Simplifying expressions is a crucial skill in many real-world applications, including:

Physics: Simplifying expressions is essential in physics to solve equations and inequalities related to motion, energy, and momentum.
Engineering: Simplifying expressions is crucial in engineering to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Simplifying expressions is essential in economics to model and analyze economic systems, such as supply and demand.

Tips and Tricks

When factoring the numerator and denominator, look for the greatest common factor (GCF) and factor it out.
Use the distributive property to expand the expression and simplify it.
Cancel out common factors by dividing both the numerator and denominator by the common factor.
Use the final simplified expression to solve equations and inequalities.

Real-World Applications

Simplifying expressions is a crucial skill in many real-world applications, including:

Physics: Simplifying expressions is essential in physics to solve equations and inequalities related to motion, energy, and momentum.
Engineering: Simplifying expressions is crucial in engineering to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Simplifying expressions is essential in economics to model and analyze economic systems, such as supply and demand.

Final Thoughts

Additional Resources

For more information on simplifying expressions, check out the following resources:

Khan Academy: Simplifying Expressions
Mathway: Simplifying Expressions
Wolfram Alpha: Simplifying Expressions

Conclusion

In this article, we answered some common questions related to simplifying expressions and provided additional tips and tricks to help you master this technique. By following these tips and tricks, you can simplify expressions and arrive at the final solution. Remember to practice regularly and apply simplifying expressions in real-world applications to become proficient in this technique.