Simplify The Expression: X 2 + 5 X ( X + 3 ) ( X + 5 ) − 4 X 2 + 12 X ( X + 3 ) ( X + 5 ) \frac{x^2+5x}{(x+3)(x+5)} - \frac{4x^2+12x}{(x+3)(x+5)} ( X + 3 ) ( X + 5 ) X 2 + 5 X ​ − ( X + 3 ) ( X + 5 ) 4 X 2 + 12 X ​

Introduction

When dealing with algebraic expressions, simplifying them is an essential step in solving equations and inequalities. In this article, we will focus on simplifying a given expression by combining two fractions. The expression we will be working with is $\frac{x^2+5x}{(x+3)(x+5)} - \frac{4x^2+12x}{(x+3)(x+5)}$. Our goal is to simplify this expression by finding a common denominator and combining the fractions.

Understanding the Expression

Before we start simplifying the expression, let's take a closer look at it. We have two fractions, each with a numerator and a denominator. The numerators are $x^2+5x$ and $4x^2+12x$, while the denominators are $(x+3)(x+5)$ for both fractions. The expression is a subtraction of the two fractions.

Finding a Common Denominator

To combine the fractions, we need to find a common denominator. In this case, the denominators are already the same, which is $(x+3)(x+5)$. This means that we can directly subtract the numerators without changing the denominators.

Simplifying the Expression

Now that we have a common denominator, we can simplify the expression by subtracting the numerators. To do this, we need to subtract the second numerator from the first numerator.

$\frac{x^2+5x}{(x+3)(x+5)} - \frac{4x^2+12x}{(x+3)(x+5)}$

$= \frac{x^2+5x - (4x^2+12x)}{(x+3)(x+5)}$

$= \frac{x^2+5x - 4x^2 - 12x}{(x+3)(x+5)}$

$= \frac{-3x^2 - 7x}{(x+3)(x+5)}$

Factoring the Numerator

Now that we have simplified the expression, we can factor the numerator to see if there are any common factors that can be canceled out with the denominator.

$= \frac{-3x(x+7/3)}{(x+3)(x+5)}$

Canceling Out Common Factors

We can see that the numerator and the denominator have a common factor of $x$. We can cancel out this common factor to simplify the expression further.

$= \frac{-3(x+7/3)}{(x+5)}$

Final Simplification

Now that we have canceled out the common factor, we can simplify the expression further by combining the terms in the numerator.

$= \frac{-3x - 7}{(x+5)}$

Conclusion

In this article, we simplified the expression $\frac{x^2+5x}{(x+3)(x+5)} - \frac{4x^2+12x}{(x+3)(x+5)}$ by finding a common denominator and combining the fractions. We then factored the numerator and canceled out common factors to simplify the expression further. The final simplified expression is $\frac{-3x - 7}{(x+5)}$.

Common Mistakes to Avoid

When simplifying expressions, there are several common mistakes to avoid. Here are a few:

• Not finding a common denominator before combining fractions
• Not factoring the numerator to see if there are any common factors that can be canceled out with the denominator
• Not canceling out common factors when they exist
• Not simplifying the expression further by combining terms in the numerator

Tips for Simplifying Expressions

Here are a few tips for simplifying expressions:

• Always find a common denominator before combining fractions
• Factor the numerator to see if there are any common factors that can be canceled out with the denominator
• Cancel out common factors when they exist
• Simplify the expression further by combining terms in the numerator
• Use algebraic properties such as the distributive property and the commutative property to simplify expressions

Real-World Applications

Simplifying expressions has many real-world applications. Here are a few:

• In physics, simplifying expressions is used to solve problems involving motion and energy.
• In engineering, simplifying expressions is used to design and optimize systems.
• In economics, simplifying expressions is used to model and analyze economic systems.

Final Thoughts

Simplifying expressions is an essential step in solving equations and inequalities. By following the steps outlined in this article, you can simplify expressions and solve problems in a variety of fields. Remember to always find a common denominator, factor the numerator, cancel out common factors, and simplify the expression further by combining terms in the numerator. With practice and patience, you can become proficient in simplifying expressions and solving problems.

Introduction

In our previous article, we simplified the expression $\frac{x^2+5x}{(x+3)(x+5)} - \frac{4x^2+12x}{(x+3)(x+5)}$ by finding a common denominator and combining the fractions. We then factored the numerator and canceled out common factors to simplify the expression further. In this article, we will answer some frequently asked questions about simplifying expressions.

Q&A

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

A: The first step in simplifying an expression is to find a common denominator. This is because you cannot combine fractions unless they have the same denominator.

Q: How do I find a common denominator?

A: To find a common denominator, you need to identify the denominators of the fractions and find the least common multiple (LCM) of the two denominators. In the case of the expression $\frac{x^2+5x}{(x+3)(x+5)} - \frac{4x^2+12x}{(x+3)(x+5)}$, the denominators are already the same, which is $(x+3)(x+5)$.

Q: What is the difference between a common denominator and a least common multiple (LCM)?

A: A common denominator is the smallest number that both denominators can divide into evenly, while a least common multiple (LCM) is the smallest number that both numbers can divide into evenly. In the case of the expression $\frac{x^2+5x}{(x+3)(x+5)} - \frac{4x^2+12x}{(x+3)(x+5)}$, the common denominator is $(x+3)(x+5)$, which is also the LCM of the two denominators.

Q: How do I factor the numerator?

A: To factor the numerator, you need to identify any common factors in the numerator and factor them out. In the case of the expression $\frac{x^2+5x}{(x+3)(x+5)} - \frac{4x^2+12x}{(x+3)(x+5)}$, the numerator can be factored as $x(x+5)$.

Q: What is the difference between factoring and canceling out common factors?

A: Factoring involves identifying any common factors in the numerator and factoring them out, while canceling out common factors involves identifying any common factors between the numerator and the denominator and canceling them out. In the case of the expression $\frac{x^2+5x}{(x+3)(x+5)} - \frac{4x^2+12x}{(x+3)(x+5)}$, we factored the numerator as $x(x+5)$ and then canceled out the common factor $x$ between the numerator and the denominator.

Q: How do I simplify an expression further?

A: To simplify an expression further, you need to combine any like terms in the numerator. In the case of the expression $\frac{x^2+5x}{(x+3)(x+5)} - \frac{4x^2+12x}{(x+3)(x+5)}$, we simplified the expression further by combining the terms in the numerator.

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

A: Some common mistakes to avoid when simplifying expressions include not finding a common denominator, not factoring the numerator, not canceling out common factors, and not simplifying the expression further by combining like terms.

Conclusion

Simplifying expressions is an essential step in solving equations and inequalities. By following the steps outlined in this article, you can simplify expressions and solve problems in a variety of fields. Remember to always find a common denominator, factor the numerator, cancel out common factors, and simplify the expression further by combining like terms. With practice and patience, you can become proficient in simplifying expressions and solving problems.

Final Thoughts

Simplifying expressions is a crucial skill in mathematics and has many real-world applications. By mastering the techniques outlined in this article, you can simplify expressions and solve problems in a variety of fields. Remember to always find a common denominator, factor the numerator, cancel out common factors, and simplify the expression further by combining like terms. With practice and patience, you can become proficient in simplifying expressions and solving problems.

Common Mistakes to Avoid

When simplifying expressions, there are several common mistakes to avoid. Here are a few:

• Not finding a common denominator before combining fractions
• Not factoring the numerator to see if there are any common factors that can be canceled out with the denominator
• Not canceling out common factors when they exist
• Not simplifying the expression further by combining like terms

Tips for Simplifying Expressions

Here are a few tips for simplifying expressions:

• Always find a common denominator before combining fractions
• Factor the numerator to see if there are any common factors that can be canceled out with the denominator
• Cancel out common factors when they exist
• Simplify the expression further by combining like terms
• Use algebraic properties such as the distributive property and the commutative property to simplify expressions

Real-World Applications

Simplifying expressions has many real-world applications. Here are a few:

• In physics, simplifying expressions is used to solve problems involving motion and energy.
• In engineering, simplifying expressions is used to design and optimize systems.
• In economics, simplifying expressions is used to model and analyze economic systems.

Final Thoughts

Simplifying expressions is a crucial skill in mathematics and has many real-world applications. By mastering the techniques outlined in this article, you can simplify expressions and solve problems in a variety of fields. Remember to always find a common denominator, factor the numerator, cancel out common factors, and simplify the expression further by combining like terms. With practice and patience, you can become proficient in simplifying expressions and solving problems.