Simplify The Expression \[$-2(p+4)^2-3+5p\$\]. What Is The Simplified Expression In Standard Form?A. \[$4p^2 + 37p - 67\$\] B. \[$-2p^2 + 13p + 13\$\] C. \[$-2p^2 - 11p - 35\$\] D. \[$2p^2 + 21p + 29\$\]

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will focus on simplifying the expression −2(p+4)2−3+5p{-2(p+4)^2-3+5p}. We will break down the process into manageable steps, using a combination of algebraic manipulations and mathematical properties to arrive at the simplified expression in standard form.

Step 1: Expand the Squared Term

The first step in simplifying the expression is to expand the squared term (p+4)2{(p+4)^2}. Using the formula (a+b)2=a2+2ab+b2{(a+b)^2 = a^2 + 2ab + b^2}, we can expand the squared term as follows:

(p+4)2=p2+2â‹…pâ‹…4+42{(p+4)^2 = p^2 + 2 \cdot p \cdot 4 + 4^2}

Simplifying the expression, we get:

(p+4)2=p2+8p+16{(p+4)^2 = p^2 + 8p + 16}

Step 2: Substitute the Expanded Term Back into the Original Expression

Now that we have expanded the squared term, we can substitute it back into the original expression:

−2(p+4)2−3+5p=−2(p2+8p+16)−3+5p{-2(p+4)^2-3+5p = -2(p^2 + 8p + 16) - 3 + 5p}

Step 3: Distribute the Negative 2 to the Terms Inside the Parentheses

To simplify the expression further, we need to distribute the negative 2 to the terms inside the parentheses:

−2(p2+8p+16)=−2p2−16p−32{-2(p^2 + 8p + 16) = -2p^2 - 16p - 32}

Substituting this back into the expression, we get:

−2p2−16p−32−3+5p{-2p^2 - 16p - 32 - 3 + 5p}

Step 4: Combine Like Terms

Now that we have distributed the negative 2, we can combine like terms:

−2p2−16p−32−3+5p=−2p2−11p−35{-2p^2 - 16p - 32 - 3 + 5p = -2p^2 - 11p - 35}

Conclusion

In conclusion, the simplified expression in standard form is −2p2−11p−35{-2p^2 - 11p - 35}. This is the correct answer among the options provided.

Answer

The correct answer is:

C. −2p2−11p−35{-2p^2 - 11p - 35}

Discussion

This problem requires a good understanding of algebraic manipulations, including expanding squared terms and distributing coefficients to terms inside parentheses. It also requires the ability to combine like terms and simplify expressions.

Tips and Tricks

  • When expanding squared terms, use the formula (a+b)2=a2+2ab+b2{(a+b)^2 = a^2 + 2ab + b^2}.
  • When distributing coefficients to terms inside parentheses, make sure to multiply each term by the coefficient.
  • When combining like terms, group terms with the same variable and exponent together.

Practice Problems

  • Simplify the expression (2x+3)2−4x+5{(2x+3)^2 - 4x + 5}.
  • Simplify the expression −3(x−2)2+2x−1{-3(x-2)^2 + 2x - 1}.

Conclusion

Introduction

In our previous article, we explored the process of simplifying algebraic expressions, focusing on the expression −2(p+4)2−3+5p{-2(p+4)^2-3+5p}. We broke down the process into manageable steps, using a combination of algebraic manipulations and mathematical properties to arrive at the simplified expression in standard form. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to identify any squared terms and expand them using the formula (a+b)2=a2+2ab+b2{(a+b)^2 = a^2 + 2ab + b^2}.

Q: How do I distribute a coefficient to terms inside parentheses?

A: To distribute a coefficient to terms inside parentheses, multiply each term by the coefficient. For example, if you have the expression −2(p+4)2{-2(p+4)^2}, you would multiply each term inside the parentheses by −2{-2}.

Q: What is the difference between combining like terms and simplifying an expression?

A: Combining like terms involves grouping terms with the same variable and exponent together and adding or subtracting their coefficients. Simplifying an expression involves using algebraic manipulations to rewrite the expression in a more compact or simplified form.

Q: Can I simplify an expression by canceling out terms?

A: No, you cannot simplify an expression by canceling out terms. Canceling out terms is not a valid algebraic manipulation and can lead to incorrect results.

Q: How do I know when an expression is simplified?

A: An expression is simplified when it cannot be rewritten in a more compact or simplified form using algebraic manipulations. In other words, an expression is simplified when it is in its simplest form.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

  • Canceling out terms
  • Forgetting to distribute coefficients to terms inside parentheses
  • Not combining like terms
  • Not using the correct order of operations

Q: Can I use a calculator to simplify algebraic expressions?

A: Yes, you can use a calculator to simplify algebraic expressions. However, it's always a good idea to double-check your work by simplifying the expression by hand.

Q: How do I know which algebraic manipulations to use when simplifying an expression?

A: The type of algebraic manipulation to use when simplifying an expression depends on the specific expression and the goal of the simplification. Some common algebraic manipulations include:

  • Expanding squared terms
  • Distributing coefficients to terms inside parentheses
  • Combining like terms
  • Using the order of operations

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By following the steps outlined in this article and avoiding common mistakes, you can simplify even the most complex expressions. Remember to expand squared terms, distribute coefficients to terms inside parentheses, and combine like terms to arrive at the simplified expression in standard form.

Practice Problems

  • Simplify the expression (2x+3)2−4x+5{(2x+3)^2 - 4x + 5}.
  • Simplify the expression −3(x−2)2+2x−1{-3(x-2)^2 + 2x - 1}.

Additional Resources

  • For more practice problems and examples, visit our website or consult a math textbook.
  • For help with specific algebraic manipulations or simplification techniques, contact a math tutor or instructor.