What Is The Correct Order Of Operations For Simplifying The Expression $2 - 3x + 5 + 7(y - 5x)^2 + X ? ? ? A. Square The Binomial, Combine Like Terms, And Then Distribute 7 Inside The Parentheses.B. Combine Like Terms, Square The Binomial, And

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What is the Correct Order of Operations for Simplifying the Expression?

Understanding the Order of Operations

When simplifying complex mathematical expressions, it's essential to follow a specific order of operations to ensure accuracy and avoid errors. The correct order of operations is a set of rules that dictate the order in which mathematical operations should be performed when there are multiple operations in an expression. In this article, we will explore the correct order of operations for simplifying the expression $2 - 3x + 5 + 7(y - 5x)^2 + x.$

The Order of Operations

The order of operations is often remembered using the acronym PEMDAS, which stands for:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Simplifying the Expression

Now that we have a clear understanding of the order of operations, let's apply it to the given expression $2 - 3x + 5 + 7(y - 5x)^2 + x.$

Step 1: Evaluate Expressions Inside Parentheses

The first step is to evaluate any expressions inside parentheses. In this case, we have the expression (y−5x)2(y - 5x)^2 inside parentheses. To evaluate this expression, we need to square the binomial.

Step 2: Square the Binomial

To square the binomial, we need to multiply the two terms inside the parentheses by each other.

(y−5x)2=(y−5x)(y−5x)(y - 5x)^2 = (y - 5x)(y - 5x)

Using the FOIL method, we can multiply the two terms as follows:

(y−5x)(y−5x)=y2−5xy−5xy+25x2(y - 5x)(y - 5x) = y^2 - 5xy - 5xy + 25x^2

Combining like terms, we get:

(y−5x)2=y2−10xy+25x2(y - 5x)^2 = y^2 - 10xy + 25x^2

Step 3: Distribute 7 Inside the Parentheses

Now that we have squared the binomial, we can distribute the 7 inside the parentheses.

7(y−5x)2=7(y2−10xy+25x2)7(y - 5x)^2 = 7(y^2 - 10xy + 25x^2)

Using the distributive property, we can multiply the 7 by each term inside the parentheses as follows:

7(y2−10xy+25x2)=7y2−70xy+175x27(y^2 - 10xy + 25x^2) = 7y^2 - 70xy + 175x^2

Step 4: Combine Like Terms

Now that we have distributed the 7 inside the parentheses, we can combine like terms.

2−3x+5+7y2−70xy+175x2+x2 - 3x + 5 + 7y^2 - 70xy + 175x^2 + x

Combining like terms, we get:

7y2−70xy+175x2+2+5−3x+x7y^2 - 70xy + 175x^2 + 2 + 5 - 3x + x

Simplifying further, we get:

7y2−70xy+175x2−2x+77y^2 - 70xy + 175x^2 - 2x + 7

Conclusion

In conclusion, the correct order of operations for simplifying the expression $2 - 3x + 5 + 7(y - 5x)^2 + x$ is to:

  1. Evaluate expressions inside parentheses.
  2. Square the binomial.
  3. Distribute 7 inside the parentheses.
  4. Combine like terms.

By following this order of operations, we can simplify the expression accurately and avoid errors.

Common Mistakes to Avoid

When simplifying complex mathematical expressions, it's essential to avoid common mistakes. Some common mistakes to avoid include:

  • Not following the order of operations.
  • Not evaluating expressions inside parentheses first.
  • Not squaring the binomial correctly.
  • Not distributing coefficients correctly.
  • Not combining like terms correctly.

By avoiding these common mistakes, we can ensure that our simplifications are accurate and reliable.

Real-World Applications

The order of operations has many real-world applications. In mathematics, the order of operations is used to simplify complex expressions and equations. In science and engineering, the order of operations is used to solve problems and make calculations. In finance and economics, the order of operations is used to calculate interest rates and investments.

Conclusion

In conclusion, the correct order of operations for simplifying the expression $2 - 3x + 5 + 7(y - 5x)^2 + x$ is to:

  1. Evaluate expressions inside parentheses.
  2. Square the binomial.
  3. Distribute 7 inside the parentheses.
  4. Combine like terms.

By following this order of operations, we can simplify the expression accurately and avoid errors. The order of operations has many real-world applications and is an essential tool for mathematicians, scientists, engineers, and finance professionals.
Frequently Asked Questions (FAQs) About the Order of Operations

Q: What is the order of operations?

A: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed when there are multiple operations in an expression. The order of operations is often remembered using the acronym PEMDAS, which stands for:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: Why is the order of operations important?

A: The order of operations is important because it ensures that mathematical expressions are evaluated accurately and consistently. Without the order of operations, mathematical expressions could be evaluated differently depending on the order in which they are performed, leading to errors and inconsistencies.

Q: What happens if I don't follow the order of operations?

A: If you don't follow the order of operations, you may get incorrect results. For example, if you have the expression $2 + 3 \times 4$ and you evaluate it from left to right, you would get $2 + 12 = 14$. However, if you follow the order of operations, you would evaluate the multiplication first, getting $3 \times 4 = 12$, and then add 2, getting $12 + 2 = 14$. In this case, the order of operations doesn't change the result, but in more complex expressions, it can make a big difference.

Q: Can I use the order of operations to simplify complex expressions?

A: Yes, the order of operations can be used to simplify complex expressions. By following the order of operations, you can break down complex expressions into smaller, more manageable parts, and then combine them to get the final result.

Q: How do I apply the order of operations to a complex expression?

A: To apply the order of operations to a complex expression, follow these steps:

  1. Evaluate any expressions inside parentheses.
  2. Evaluate any exponential expressions.
  3. Evaluate any multiplication and division operations from left to right.
  4. Evaluate any addition and subtraction operations from left to right.

Q: What are some common mistakes to avoid when using the order of operations?

A: Some common mistakes to avoid when using the order of operations include:

  • Not following the order of operations.
  • Not evaluating expressions inside parentheses first.
  • Not squaring the binomial correctly.
  • Not distributing coefficients correctly.
  • Not combining like terms correctly.

Q: Can I use the order of operations to solve equations?

A: Yes, the order of operations can be used to solve equations. By following the order of operations, you can simplify complex equations and solve for the unknown variable.

Q: How do I use the order of operations to solve equations?

A: To use the order of operations to solve equations, follow these steps:

  1. Simplify the equation by combining like terms.
  2. Evaluate any expressions inside parentheses.
  3. Evaluate any exponential expressions.
  4. Evaluate any multiplication and division operations from left to right.
  5. Evaluate any addition and subtraction operations from left to right.

Q: What are some real-world applications of the order of operations?

A: The order of operations has many real-world applications, including:

  • Mathematics: The order of operations is used to simplify complex expressions and equations.
  • Science and engineering: The order of operations is used to solve problems and make calculations.
  • Finance and economics: The order of operations is used to calculate interest rates and investments.

Conclusion

In conclusion, the order of operations is a set of rules that dictate the order in which mathematical operations should be performed when there are multiple operations in an expression. By following the order of operations, you can simplify complex expressions and equations, and solve for the unknown variable. Remember to avoid common mistakes, such as not following the order of operations, and to use the order of operations to solve equations and make calculations.