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 students and professionals alike. 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+2(p)(4)+42{(p+4)^2 = p^2 + 2(p)(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 Squared Term

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

The next step is to distribute the negative 2 to each term inside the parentheses:

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

Step 4: Combine Like Terms

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

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

Combining the like terms, we get:

−2p2−11p−35{-2p^2 - 11p - 35}

Conclusion

In conclusion, the simplified expression in standard form is −2p2−11p−35{-2p^2 - 11p - 35}. This expression is in the standard form of a quadratic equation, which is ax2+bx+c{ax^2 + bx + c}, where a{a}, b{b}, and c{c} are constants.

Answer

The correct answer is:

  • C. −2p2−11p−35{-2p^2 - 11p - 35}*

This answer is the result of simplifying the expression −2(p+4)2−3+5p{-2(p+4)^2-3+5p} using a combination of algebraic manipulations and mathematical properties.

Discussion

Simplifying algebraic expressions is an essential skill for students and professionals alike. In this article, we have demonstrated a step-by-step approach to simplifying the expression −2(p+4)2−3+5p{-2(p+4)^2-3+5p}. By following these steps, you can simplify any algebraic expression and arrive at the simplified expression in standard form.

Tips and Tricks

Here are some tips and tricks to help you simplify algebraic expressions:

  • Use the distributive property: The distributive property states that for any real numbers a{a}, b{b}, and c{c}, a(b+c)=ab+ac{a(b+c) = ab + ac}. This property can be used to simplify expressions by distributing the terms inside the parentheses.
  • Combine like terms: Like terms are terms that have the same variable and exponent. Combining like terms can help simplify expressions by eliminating unnecessary terms.
  • Use the order of operations: The order of operations states that expressions should be evaluated in the following order: parentheses, exponents, multiplication and division, and addition and subtraction. This order can help simplify expressions by ensuring that the correct operations are performed in the correct order.

Practice Problems

Here are some practice problems to help you practice simplifying algebraic expressions:

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

Introduction

Simplifying algebraic expressions is an essential skill for students and professionals alike. In our previous article, we demonstrated a step-by-step approach to simplifying the expression −2(p+4)2−3+5p{-2(p+4)^2-3+5p}. In this article, we will answer some frequently asked questions (FAQs) 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 expand any squared terms. This involves using the formula (a+b)2=a2+2ab+b2{(a+b)^2 = a^2 + 2ab + b^2} to expand the squared term.

Q: How do I distribute a negative number to a term inside the parentheses?

A: To distribute a negative number to a term inside the parentheses, you need to multiply the negative number by each term inside the parentheses. For example, if you have the expression −2(x+3){-2(x+3)}, you would distribute the negative 2 to each term inside the parentheses as follows:

−2(x+3)=−2x−6{-2(x+3) = -2x - 6}

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable and exponent. Unlike terms are terms that have different variables or exponents. For example, the terms 2x{2x} and 3x{3x} are like terms, while the terms 2x{2x} and 5y{5y} are unlike terms.

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, if you have the expression 2x+3x{2x + 3x}, you would combine the like terms as follows:

2x+3x=5x{2x + 3x = 5x}

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. The order of operations is as follows:

  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: How do I simplify an expression with multiple variables?

A: To simplify an expression with multiple variables, you need to follow the same steps as before. However, you may need to use the distributive property and combine like terms more frequently.

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

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

  • Forgetting to distribute negative numbers to terms inside the parentheses.
  • Failing to combine like terms.
  • Not following the order of operations.
  • Making errors when expanding squared terms.

Conclusion

Simplifying algebraic expressions is an essential skill for students and professionals alike. By following the steps outlined in this article and practicing with these FAQs, you can become proficient in simplifying algebraic expressions and arrive at the simplified expression in standard form.

Practice Problems

Here are some practice problems to help you practice simplifying algebraic expressions:

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

By following the steps outlined in this article and practicing with these practice problems, you can become proficient in simplifying algebraic expressions and arrive at the simplified expression in standard form.