Select All The Correct Answers.Which Expressions Are Equivalent To The Expression 30 X 2 − 5 X − 10 30x^2 - 5x - 10 30 X 2 − 5 X − 10 ?A. − 5 ( − 6 X 2 + X + 2 -5(-6x^2 + X + 2 − 5 ( − 6 X 2 + X + 2 ] B. 5 X ( 6 X − X − 2 5x(6x - X - 2 5 X ( 6 X − X − 2 ] C. 3 X ( 2 X − 1 ) + 2 ( 2 X − 1 3x(2x - 1) + 2(2x - 1 3 X ( 2 X − 1 ) + 2 ( 2 X − 1 ] D. ( 10 X − 5 ) ( 3 X − 5 (10x - 5)(3x - 5 ( 10 X − 5 ) ( 3 X − 5 ] E.

Introduction

In algebra, equivalent expressions are those that have the same value for all possible values of the variable. In this article, we will explore the concept of equivalent expressions and analyze the given options to determine which ones are equivalent to the expression $30x^2 - 5x - 10$.

Understanding Equivalent Expressions

Equivalent expressions are expressions that have the same value for all possible values of the variable. This means that if we substitute any value of the variable into two equivalent expressions, the results will be the same. For example, the expressions $2x + 3$ and $x + 2 + x + 1$ are equivalent because they both simplify to $x + 3$.

Factoring the Given Expression

To determine which expressions are equivalent to the given expression $30x^2 - 5x - 10$, we need to factor it. Factoring an expression involves expressing it as a product of simpler expressions. In this case, we can factor the given expression as follows:

$30x^2 - 5x - 10 = 5(6x^2 - x - 2)$

Analyzing the Options

Now that we have factored the given expression, we can analyze the options to determine which ones are equivalent.

Option A: $-5(-6x^2 + x + 2)$

To determine if this expression is equivalent, we need to simplify it.

$-5(-6x^2 + x + 2) = 30x^2 - 5x - 10$

This expression is equivalent to the given expression.

Option B: $5x(6x - x - 2)$

To determine if this expression is equivalent, we need to simplify it.

$5x(6x - x - 2) = 30x^2 - 5x - 10$

This expression is equivalent to the given expression.

Option C: $3x(2x - 1) + 2(2x - 1)$

To determine if this expression is equivalent, we need to simplify it.

$3x(2x - 1) + 2(2x - 1) = 6x^2 - 3x + 4x - 2 = 6x^2 + x - 2$

This expression is not equivalent to the given expression.

Option D: $(10x - 5)(3x - 5)$

To determine if this expression is equivalent, we need to simplify it.

$(10x - 5)(3x - 5) = 30x^2 - 50x - 15x + 25 = 30x^2 - 65x + 25$

This expression is not equivalent to the given expression.

Conclusion

In conclusion, the expressions $-5(-6x^2 + x + 2)$ and $5x(6x - x - 2)$ are equivalent to the expression $30x^2 - 5x - 10$. These expressions can be obtained by factoring the given expression and simplifying the resulting expression.

Key Takeaways

• Equivalent expressions are those that have the same value for all possible values of the variable.
• Factoring an expression involves expressing it as a product of simpler expressions.
• To determine if an expression is equivalent, we need to simplify it and compare it to the given expression.

Final Answer

The final answer is:

• A. $-5(-6x^2 + x + 2)$
• B. $5x(6x - x - 2)$
  

Introduction

In our previous article, we explored the concept of equivalent expressions and analyzed the given options to determine which ones are equivalent to the expression $30x^2 - 5x - 10$. In this article, we will provide a comprehensive Q&A guide to help you understand equivalent expressions and how to work with them.

Q: What are equivalent expressions?

A: Equivalent expressions are expressions that have the same value for all possible values of the variable. This means that if we substitute any value of the variable into two equivalent expressions, the results will be the same.

Q: How do I determine if two expressions are equivalent?

A: To determine if two expressions are equivalent, we need to simplify them and compare them. We can simplify an expression by combining like terms, factoring, or using other algebraic techniques.

Q: What is factoring?

A: Factoring an expression involves expressing it as a product of simpler expressions. For example, the expression $6x^2 + 12x$ can be factored as $6x(x + 2)$.

Q: How do I factor an expression?

A: To factor an expression, we need to identify the greatest common factor (GCF) of the terms and then express the expression as a product of the GCF and the remaining terms.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) of a set of numbers is the largest number that divides each of the numbers without leaving a remainder. For example, the GCF of 12 and 18 is 6.

Q: How do I simplify an expression?

A: To simplify an expression, we need to combine like terms, eliminate any parentheses, and perform any necessary operations.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, the terms $3x^2$ and $5x^2$ are like terms because they both have the variable $x$ and the exponent $2$.

Q: How do I combine like terms?

A: To combine like terms, we need to add or subtract the coefficients of the like terms. For example, the expression $3x^2 + 5x^2$ can be simplified to $8x^2$ by combining the like terms.

Q: What is the difference between equivalent expressions and equivalent equations?

A: Equivalent expressions are expressions that have the same value for all possible values of the variable. Equivalent equations, on the other hand, are equations that have the same solution set.

Q: How do I determine if two equations are equivalent?

A: To determine if two equations are equivalent, we need to compare their solution sets. If the solution sets are the same, then the equations are equivalent.

Q: What is the importance of equivalent expressions in algebra?

A: Equivalent expressions are important in algebra because they allow us to simplify complex expressions and solve equations more easily. By expressing an expression in a simpler form, we can make it easier to work with and solve.

Q: How do I apply equivalent expressions in real-world problems?

A: Equivalent expressions can be applied in a variety of real-world problems, such as finance, science, and engineering. For example, in finance, equivalent expressions can be used to calculate interest rates and investment returns.

Conclusion

In conclusion, equivalent expressions are an important concept in algebra that allows us to simplify complex expressions and solve equations more easily. By understanding equivalent expressions and how to work with them, we can apply them in a variety of real-world problems.

Key Takeaways

• Equivalent expressions are expressions that have the same value for all possible values of the variable.
• Factoring an expression involves expressing it as a product of simpler expressions.
• To determine if two expressions are equivalent, we need to simplify them and compare them.
• Equivalent expressions can be applied in a variety of real-world problems, such as finance, science, and engineering.

Final Answer

The final answer is:

• Equivalent expressions are expressions that have the same value for all possible values of the variable.
• Factoring an expression involves expressing it as a product of simpler expressions.
• To determine if two expressions are equivalent, we need to simplify them and compare them.
• Equivalent expressions can be applied in a variety of real-world problems, such as finance, science, and engineering.