What Is An Equivalent Expression Of X 2 + 3 X − 18 7 X − 21 \frac{x^2+3x-18}{7x-21} 7 X − 21 X 2 + 3 X − 18 ​ In Lowest Terms?$\square$2. What Values Of X X X Must We Exclude From The Domains Of The Expressions?

$$

===========================================================

Introduction

---

In mathematics, equivalent expressions are expressions that have the same value for all possible input values. In this article, we will explore how to find an equivalent expression of $\frac{x^2+3x-18}{7x-21}$ in lowest terms. We will also discuss the values of $x$ that must be excluded from the domains of the expressions.

Understanding the Problem

---

The given expression is a rational expression, which is a fraction of two polynomials. To find an equivalent expression in lowest terms, we need to simplify the fraction by canceling out any common factors in the numerator and denominator.

Simplifying the Expression

---

Let's start by factoring the numerator and denominator:

$$\frac{x^2+3x-18}{7x-21} = \frac{(x+6)(x-3)}{7(x-3)}$$

We can see that the numerator and denominator have a common factor of $(x-3)$. We can cancel out this factor to get:

$$\frac{x^2+3x-18}{7x-21} = \frac{x+6}{7}$$

This is the equivalent expression in lowest terms.

Excluding Values from the Domain

---

When we simplified the expression, we canceled out the factor $(x-3)$. This means that the value $x=3$ must be excluded from the domain of the expression. This is because when $x=3$, the denominator of the original expression would be zero, which would make the expression undefined.

Why Excluding Values is Important

---

Excluding values from the domain of an expression is important because it ensures that the expression is well-defined and can be evaluated for all possible input values. If we don't exclude values from the domain, we may end up with an expression that is undefined or has multiple values.

Example: Finding Equivalent Expressions

---

Let's consider another example of finding equivalent expressions. Suppose we have the expression $\frac{x^2-4x+4}{x-2}$. We can simplify this expression by factoring the numerator and denominator:

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

We can cancel out the factor $(x-2)$ to get:

$$\frac{x^2-4x+4}{x-2} = x-2$$

This is the equivalent expression in lowest terms.

Conclusion

---

In this article, we explored how to find an equivalent expression of $\frac{x^2+3x-18}{7x-21}$ in lowest terms. We also discussed the values of $x$ that must be excluded from the domains of the expressions. By following these steps, we can simplify rational expressions and ensure that they are well-defined for all possible input values.

Frequently Asked Questions

---

• Q: What is an equivalent expression? A: An equivalent expression is an expression that has the same value for all possible input values.
• Q: How do I simplify a rational expression? A: To simplify a rational expression, you can factor the numerator and denominator and cancel out any common factors.
• Q: Why do I need to exclude values from the domain? A: Excluding values from the domain ensures that the expression is well-defined and can be evaluated for all possible input values.

Final Thoughts

---

Finding equivalent expressions is an important skill in mathematics, and it has many real-world applications. By following the steps outlined in this article, you can simplify rational expressions and ensure that they are well-defined for all possible input values.

===========================================================

Introduction

---

In our previous article, we explored how to find an equivalent expression of $\frac{x^2+3x-18}{7x-21}$ in lowest terms. We also discussed the values of $x$ that must be excluded from the domains of the expressions. In this article, we will answer some frequently asked questions about equivalent expressions.

Q&A

---

Q: What is an equivalent expression?

A: An equivalent expression is an expression that has the same value for all possible input values.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you can factor the numerator and denominator and cancel out any common factors.

Q: Why do I need to exclude values from the domain?

A: Excluding values from the domain ensures that the expression is well-defined and can be evaluated for all possible input values.

Q: What is the difference between an equivalent expression and a simplified expression?

A: An equivalent expression is an expression that has the same value for all possible input values, while a simplified expression is an expression that has been reduced to its simplest form.

Q: Can I simplify an expression that has a variable in the denominator?

A: Yes, you can simplify an expression that has a variable in the denominator. However, you must be careful not to divide by zero.

Q: How do I know if an expression is in its simplest form?

A: An expression is in its simplest form if it cannot be simplified further by factoring or canceling out common factors.

Q: Can I use a calculator to simplify an expression?

A: Yes, you can use a calculator to simplify an expression. However, you should always check your work to make sure that the expression is in its simplest form.

Q: What is the importance of equivalent expressions in real-world applications?

A: Equivalent expressions are important in real-world applications because they allow us to simplify complex expressions and make them easier to work with.

Q: Can I use equivalent expressions to solve equations?

A: Yes, you can use equivalent expressions to solve equations. By simplifying the expressions, you can make it easier to solve the equation.

Q: How do I know if an expression is an equivalent expression?

A: An expression is an equivalent expression if it has the same value for all possible input values.

Conclusion

---

In this article, we answered some frequently asked questions about equivalent expressions. We hope that this article has been helpful in clarifying any confusion you may have had about equivalent expressions.

Final Thoughts

---

Equivalent expressions are an important concept in mathematics, and they have many real-world applications. By understanding how to simplify expressions and find equivalent expressions, you can make it easier to work with complex expressions and solve equations.

Additional Resources

---

• Mathway: A website that can help you solve math problems and simplify expressions.
• Khan Academy: A website that provides free math lessons and exercises.
• Wolfram Alpha: A website that can help you simplify expressions and solve equations.

Glossary

---

• Equivalent expression: An expression that has the same value for all possible input values.
• Simplified expression: An expression that has been reduced to its simplest form.
• Domain: The set of all possible input values for an expression.
• Variable: A symbol that represents a value that can change.
• Denominator: The bottom part of a fraction.
• Numerator: The top part of a fraction.