Simplify The Following Expression Completely: X 2 − 6 X − 27 X 2 − 4 X − 45 \frac{x^2 - 6x - 27}{x^2 - 4x - 45} X 2 − 4 X − 45 X 2 − 6 X − 27 ​ Enter The Numerator And Denominator Separately In The Boxes Below. If The Denominator Is 1, Enter The Number 1. Do Not Leave Either Box

Introduction

In this article, we will simplify the given expression completely by factoring the numerator and denominator, and then canceling out any common factors. This process will help us to simplify the expression and make it easier to work with.

Step 1: Factor the Numerator

The numerator of the given expression is $x^2 - 6x - 27$. To factor this expression, we need to find two numbers whose product is $-27$ and whose sum is $-6$. These numbers are $-9$ and $3$, so we can write the numerator as:

$$x^2 - 6x - 27 = (x - 9)(x + 3)$$

Step 2: Factor the Denominator

The denominator of the given expression is $x^2 - 4x - 45$. To factor this expression, we need to find two numbers whose product is $-45$ and whose sum is $-4$. These numbers are $-9$ and $5$, so we can write the denominator as:

$$x^2 - 4x - 45 = (x - 9)(x + 5)$$

Step 3: Simplify the Expression

Now that we have factored the numerator and denominator, we can simplify the expression by canceling out any common factors. In this case, we can cancel out the factor $(x - 9)$ from both the numerator and denominator:

$$\frac{(x - 9)(x + 3)}{(x - 9)(x + 5)} = \frac{x + 3}{x + 5}$$

Conclusion

In this article, we simplified the given expression completely by factoring the numerator and denominator, and then canceling out any common factors. The simplified expression is $\frac{x + 3}{x + 5}$.

Why is Factoring Important?

Factoring is an important concept in algebra because it allows us to simplify complex expressions and make them easier to work with. By factoring an expression, we can identify any common factors and cancel them out, which can make the expression much simpler. This is especially important in algebra because it allows us to solve equations and inequalities more easily.

Real-World Applications of Factoring

Factoring has many real-world applications in fields such as engineering, economics, and computer science. For example, in engineering, factoring is used to simplify complex mathematical models and make them easier to analyze. In economics, factoring is used to analyze the relationships between different economic variables and make predictions about future trends. In computer science, factoring is used to develop algorithms and data structures that can efficiently solve complex problems.

Common Mistakes to Avoid

When factoring an expression, there are several common mistakes to avoid. One mistake is to forget to check for common factors before canceling them out. Another mistake is to factor an expression incorrectly, which can lead to incorrect solutions. Finally, a common mistake is to forget to simplify the expression after factoring, which can make it more difficult to work with.

Tips and Tricks

When factoring an expression, there are several tips and tricks that can help. One tip is to look for common factors such as $x^2$, $x$, or constants. Another tip is to use the distributive property to expand the expression and make it easier to factor. Finally, a tip is to use a calculator or computer program to check your work and make sure that you have factored the expression correctly.

Conclusion

In conclusion, factoring is an important concept in algebra that allows us to simplify complex expressions and make them easier to work with. By factoring an expression, we can identify any common factors and cancel them out, which can make the expression much simpler. This is especially important in algebra because it allows us to solve equations and inequalities more easily. By following the tips and tricks outlined in this article, you can become more proficient in factoring and simplify complex expressions with ease.

Final Answer

Introduction

In our previous article, we simplified the given expression completely by factoring the numerator and denominator, and then canceling out any common factors. In this article, we will answer some common questions that students may have when simplifying expressions.

Q: What is factoring?

A: Factoring is the process of expressing an expression as a product of simpler expressions. In other words, it is the process of breaking down an expression into its constituent parts.

Q: Why is factoring important?

A: Factoring is important because it allows us to simplify complex expressions and make them easier to work with. By factoring an expression, we can identify any common factors and cancel them out, which can make the expression much simpler.

Q: How do I factor an expression?

A: To factor an expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the middle term. For example, to factor the expression $x^2 + 5x + 6$, you need to find two numbers whose product is $6$ and whose sum is $5$. These numbers are $2$ and $3$, so you can write the expression as $(x + 2)(x + 3)$.

Q: What are some common mistakes to avoid when factoring?

A: Some common mistakes to avoid when factoring include:

• Forgetting to check for common factors before canceling them out
• Factoring an expression incorrectly, which can lead to incorrect solutions
• Forgetting to simplify the expression after factoring, which can make it more difficult to work with

Q: How do I know if an expression can be factored?

A: To determine if an expression can be factored, you need to look for common factors such as $x^2$, $x$, or constants. If you can find any common factors, you can factor the expression.

Q: What are some tips and tricks for factoring?

A: Some tips and tricks for factoring include:

• Looking for common factors such as $x^2$, $x$, or constants
• Using the distributive property to expand the expression and make it easier to factor
• Using a calculator or computer program to check your work and make sure that you have factored the expression correctly

Q: Can I use a calculator or computer program to factor an expression?

A: Yes, you can use a calculator or computer program to factor an expression. In fact, many calculators and computer programs have built-in factoring tools that can help you factor an expression quickly and easily.

Q: How do I check my work when factoring?

A: To check your work when factoring, you need to plug in a value for the variable and see if the expression simplifies to the correct value. For example, if you factor the expression $x^2 + 5x + 6$ as $(x + 2)(x + 3)$, you can plug in $x = 1$ and see if the expression simplifies to $8$.

Conclusion

In conclusion, factoring is an important concept in algebra that allows us to simplify complex expressions and make them easier to work with. By following the tips and tricks outlined in this article, you can become more proficient in factoring and simplify complex expressions with ease.

Final Answer

The final answer is $\boxed{\frac{x + 3}{x + 5}}$.