Which Expression Is Equivalent To $2x^2 - 2x + 7$?A. ( 4 X + 12 ) + ( 2 X 2 − 6 X + 5 (4x + 12) + (2x^2 - 6x + 5 ( 4 X + 12 ) + ( 2 X 2 − 6 X + 5 ]B. ( X 2 − 5 X + 13 ) + ( X 2 + 3 X − 6 (x^2 - 5x + 13) + (x^2 + 3x - 6 ( X 2 − 5 X + 13 ) + ( X 2 + 3 X − 6 ]C. ( 4 X 2 − 6 X + 11 ) + ( 2 X 2 − 4 X + 4 (4x^2 - 6x + 11) + (2x^2 - 4x + 4 ( 4 X 2 − 6 X + 11 ) + ( 2 X 2 − 4 X + 4 ]D. ( 5 X 2 − 8 X + 120 ) + ( − 3 X 2 + 10 X − 13 (5x^2 - 8x + 120) + (-3x^2 + 10x - 13 ( 5 X 2 − 8 X + 120 ) + ( − 3 X 2 + 10 X − 13 ]

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Understanding the Problem

When dealing with algebraic expressions, it's essential to understand the concept of equivalence. Two expressions are equivalent if they have the same value for all possible values of the variable. In this case, we're given an expression $2x^2 - 2x + 7$ and asked to find an equivalent expression from the given options.

Analyzing the Options

To determine which expression is equivalent to $2x^2 - 2x + 7$, we need to carefully analyze each option. Let's start by simplifying each expression and then compare them to the given expression.

Option A: $(4x + 12) + (2x^2 - 6x + 5)$

When we simplify this expression, we get:

$(4x + 12) + (2x^2 - 6x + 5)$ $= 4x + 12 + 2x^2 - 6x + 5$ $= 2x^2 - 2x + 17$

This expression is not equivalent to $2x^2 - 2x + 7$ because the constant term is different.

Option B: $(x^2 - 5x + 13) + (x^2 + 3x - 6)$

When we simplify this expression, we get:

$(x^2 - 5x + 13) + (x^2 + 3x - 6)$ $= x^2 - 5x + 13 + x^2 + 3x - 6$ $= 2x^2 - 2x + 7$

This expression is equivalent to $2x^2 - 2x + 7$.

Option C: $(4x^2 - 6x + 11) + (2x^2 - 4x + 4)$

When we simplify this expression, we get:

$(4x^2 - 6x + 11) + (2x^2 - 4x + 4)$ $= 4x^2 - 6x + 11 + 2x^2 - 4x + 4$ $= 6x^2 - 10x + 15$

This expression is not equivalent to $2x^2 - 2x + 7$ because the coefficients of the terms are different.

Option D: $(5x^2 - 8x + 120) + (-3x^2 + 10x - 13)$

When we simplify this expression, we get:

$(5x^2 - 8x + 120) + (-3x^2 + 10x - 13)$ $= 5x^2 - 8x + 120 - 3x^2 + 10x - 13$ $= 2x^2 + 2x + 107$

This expression is not equivalent to $2x^2 - 2x + 7$ because the coefficients of the terms are different.

Conclusion

Based on our analysis, we can conclude that the expression equivalent to $2x^2 - 2x + 7$ is:

$(x^2 - 5x + 13) + (x^2 + 3x - 6)$

This expression simplifies to $2x^2 - 2x + 7$, making it the correct answer.

Tips and Tricks

When dealing with algebraic expressions, it's essential to carefully analyze each option and simplify the expressions to determine which one is equivalent to the given expression. This requires a good understanding of algebraic concepts, such as combining like terms and simplifying expressions.

Real-World Applications

Understanding equivalent expressions is crucial in various real-world applications, such as:

• Science and Engineering: Equivalent expressions are used to simplify complex mathematical models and equations, making it easier to analyze and solve problems.
• Computer Science: Equivalent expressions are used in programming to simplify complex algorithms and equations, making it easier to write efficient code.
• Finance: Equivalent expressions are used to simplify complex financial models and equations, making it easier to analyze and solve problems related to investments and risk management.

Common Mistakes

When dealing with equivalent expressions, it's common to make mistakes such as:

• Not simplifying the expressions: Failing to simplify the expressions can lead to incorrect answers.
• Not combining like terms: Failing to combine like terms can lead to incorrect answers.
• Not checking the constant term: Failing to check the constant term can lead to incorrect answers.

Conclusion

In conclusion, understanding equivalent expressions is crucial in mathematics and has various real-world applications. By carefully analyzing each option and simplifying the expressions, we can determine which expression is equivalent to the given expression.

Q: What is an equivalent expression?

A: An equivalent expression is an expression that has the same value as another expression for all possible values of the variable.

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

A: To determine if two expressions are equivalent, you need to simplify both expressions and compare them. If the simplified expressions are the same, then the original expressions are equivalent.

Q: What are some common mistakes to avoid when working with equivalent expressions?

A: Some common mistakes to avoid when working with equivalent expressions include:

• Not simplifying the expressions
• Not combining like terms
• Not checking the constant term
• Not using the correct order of operations

Q: How do I simplify an expression?

A: To simplify an expression, you need to combine like terms and eliminate any unnecessary parentheses or brackets.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, 2x and 4x are like terms because they both have the variable x and the exponent 1.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, 2x + 4x = 6x.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when working with expressions that involve multiple operations. The order of operations is:

1. Parentheses
2. Exponents
3. Multiplication and Division
4. Addition and Subtraction

Q: How do I use the order of operations to simplify an expression?

A: To use the order of operations to simplify an expression, you need to follow the order of operations and perform the operations in the correct order.

Q: What are some real-world applications of equivalent expressions?

A: Equivalent expressions have many real-world applications, including:

• Science and Engineering: Equivalent expressions are used to simplify complex mathematical models and equations, making it easier to analyze and solve problems.
• Computer Science: Equivalent expressions are used in programming to simplify complex algorithms and equations, making it easier to write efficient code.
• Finance: Equivalent expressions are used to simplify complex financial models and equations, making it easier to analyze and solve problems related to investments and risk management.

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

A: To know if an expression is equivalent to another expression, you need to simplify both expressions and compare them. If the simplified expressions are the same, then the original expressions are equivalent.

Q: What are some common pitfalls to avoid when working with equivalent expressions?

A: Some common pitfalls to avoid when working with equivalent expressions include:

• Not simplifying the expressions
• Not combining like terms
• Not checking the constant term
• Not using the correct order of operations

Q: How do I check if an expression is equivalent to another expression?

A: To check if an expression is equivalent to another expression, you need to simplify both expressions and compare them. If the simplified expressions are the same, then the original expressions are equivalent.

Q: What are some tips for working with equivalent expressions?

A: Some tips for working with equivalent expressions include:

• Simplify the expressions as much as possible
• Combine like terms
• Check the constant term
• Use the correct order of operations

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

A: To apply equivalent expressions in real-world scenarios, you need to identify the equivalent expressions and use them to simplify complex mathematical models and equations, making it easier to analyze and solve problems.

Q: What are some examples of equivalent expressions in real-world scenarios?

A: Some examples of equivalent expressions in real-world scenarios include:

• Simplifying complex financial models and equations to analyze and solve problems related to investments and risk management
• Simplifying complex algorithms and equations to write efficient code in programming
• Simplifying complex mathematical models and equations to analyze and solve problems in science and engineering

Q: How do I know if an expression is equivalent to another expression in a real-world scenario?

A: To know if an expression is equivalent to another expression in a real-world scenario, you need to simplify both expressions and compare them. If the simplified expressions are the same, then the original expressions are equivalent.

Q: What are some common challenges when working with equivalent expressions in real-world scenarios?

A: Some common challenges when working with equivalent expressions in real-world scenarios include:

• Simplifying complex expressions
• Combining like terms
• Checking the constant term
• Using the correct order of operations

Q: How do I overcome common challenges when working with equivalent expressions in real-world scenarios?

A: To overcome common challenges when working with equivalent expressions in real-world scenarios, you need to:

• Simplify the expressions as much as possible
• Combine like terms
• Check the constant term
• Use the correct order of operations

Q: What are some best practices for working with equivalent expressions in real-world scenarios?

A: Some best practices for working with equivalent expressions in real-world scenarios include:

• Simplify the expressions as much as possible
• Combine like terms
• Check the constant term
• Use the correct order of operations

Q: How do I apply best practices when working with equivalent expressions in real-world scenarios?

A: To apply best practices when working with equivalent expressions in real-world scenarios, you need to:

• Simplify the expressions as much as possible
• Combine like terms
• Check the constant term
• Use the correct order of operations

Q: What are some common mistakes to avoid when working with equivalent expressions in real-world scenarios?

A: Some common mistakes to avoid when working with equivalent expressions in real-world scenarios include:

• Not simplifying the expressions
• Not combining like terms
• Not checking the constant term
• Not using the correct order of operations

Q: How do I know if an expression is equivalent to another expression in a real-world scenario?

A: To know if an expression is equivalent to another expression in a real-world scenario, you need to simplify both expressions and compare them. If the simplified expressions are the same, then the original expressions are equivalent.

Q: What are some examples of equivalent expressions in real-world scenarios?

A: Some examples of equivalent expressions in real-world scenarios include:

• Simplifying complex financial models and equations to analyze and solve problems related to investments and risk management
• Simplifying complex algorithms and equations to write efficient code in programming
• Simplifying complex mathematical models and equations to analyze and solve problems in science and engineering

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

A: To apply equivalent expressions in real-world scenarios, you need to identify the equivalent expressions and use them to simplify complex mathematical models and equations, making it easier to analyze and solve problems.

Q: What are some tips for working with equivalent expressions in real-world scenarios?

A: Some tips for working with equivalent expressions in real-world scenarios include:

• Simplify the expressions as much as possible
• Combine like terms
• Check the constant term
• Use the correct order of operations

Q: How do I know if an expression is equivalent to another expression in a real-world scenario?

A: To know if an expression is equivalent to another expression in a real-world scenario, you need to simplify both expressions and compare them. If the simplified expressions are the same, then the original expressions are equivalent.

Q: What are some common challenges when working with equivalent expressions in real-world scenarios?

A: Some common challenges when working with equivalent expressions in real-world scenarios include:

• Simplifying complex expressions
• Combining like terms
• Checking the constant term
• Using the correct order of operations

Q: How do I overcome common challenges when working with equivalent expressions in real-world scenarios?

A: To overcome common challenges when working with equivalent expressions in real-world scenarios, you need to:

• Simplify the expressions as much as possible
• Combine like terms
• Check the constant term
• Use the correct order of operations

Q: What are some best practices for working with equivalent expressions in real-world scenarios?

A: Some best practices for working with equivalent expressions in real-world scenarios include:

• Simplify the expressions as much as possible
• Combine like terms
• Check the constant term
• Use the correct order of operations

Q: How do I apply best practices when working with equivalent expressions in real-world scenarios?

A: To apply best practices when working with equivalent expressions in real-world scenarios, you need to:

• Simplify the expressions as much as possible
• Combine like terms
• Check the constant term
• Use the correct order of operations

Q: What are some common mistakes to avoid when working with equivalent expressions in real-world scenarios?

A: Some common mistakes to avoid when working with equivalent expressions in real-world scenarios include:

• Not simplifying the expressions
• Not combining like terms
• Not checking the constant term
• Not using the correct order of operations

Q: How do I know if an expression is equivalent to another expression in a real-world scenario?

A: To know if an expression is equivalent to another expression in a real-world scenario, you need to simplify both expressions and compare them. If the simplified expressions are the same, then the original expressions are equivalent.

Q: What are some examples of equivalent expressions in real-world scenarios?

A: Some examples of equivalent expressions in real-world scenarios include:

• Simpl