Which Expression Is Equivalent To $n^2 + 26n + 88$ For All Values Of $n$?A. \[$(n+8)(n+11)\$\]B. \[$(n+4)(n+22)\$\]C. \[$(n+4)(n+24)\$\]D. \[$(n+8)(n+18)\$\]

Introduction

Quadratic expressions are a fundamental concept in algebra, and understanding how to manipulate them is crucial for solving various mathematical problems. In this article, we will explore how to find an equivalent expression for a given quadratic expression, specifically $n^2 + 26n + 88$. We will examine each option and determine which one is equivalent to the given expression.

Understanding Quadratic Expressions

A quadratic expression is a polynomial of degree two, which means it has a highest power of two. The general form of a quadratic expression is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. In our case, the given expression is $n^2 + 26n + 88$, where $a = 1$, $b = 26$, and $c = 88$.

Factoring Quadratic Expressions

One way to simplify a quadratic expression is to factor it. Factoring involves expressing the quadratic expression as a product of two binomials. The general form of a factored quadratic expression is $(x + p)(x + q)$, where $p$ and $q$ are constants. To factor a quadratic expression, we need to find two numbers whose product is equal to the constant term ($c$) and whose sum is equal to the coefficient of the linear term ($b$).

Option A: $(n+8)(n+11)$

Let's examine the first option, $(n+8)(n+11)$. To determine if this is equivalent to the given expression, we need to multiply the two binomials.

${(n+8)(n+11) = n^2 + 11n + 8n + 88}$

Combining like terms, we get:

${n^2 + 19n + 88}$

This is not equivalent to the given expression, as the coefficient of the linear term is different.

Option B: $(n+4)(n+22)$

Next, let's examine the second option, $(n+4)(n+22)$. To determine if this is equivalent to the given expression, we need to multiply the two binomials.

${(n+4)(n+22) = n^2 + 22n + 4n + 88}$

Combining like terms, we get:

${n^2 + 26n + 88}$

This is equivalent to the given expression, as the coefficients of the linear and constant terms are the same.

Option C: $(n+4)(n+24)$

Let's examine the third option, $(n+4)(n+24)$. To determine if this is equivalent to the given expression, we need to multiply the two binomials.

${(n+4)(n+24) = n^2 + 24n + 4n + 96}$

Combining like terms, we get:

${n^2 + 28n + 96}$

This is not equivalent to the given expression, as the coefficient of the linear term is different.

Option D: $(n+8)(n+18)$

Finally, let's examine the fourth option, $(n+8)(n+18)$. To determine if this is equivalent to the given expression, we need to multiply the two binomials.

${(n+8)(n+18) = n^2 + 18n + 8n + 144}$

Combining like terms, we get:

${n^2 + 26n + 144}$

This is not equivalent to the given expression, as the constant term is different.

Conclusion

In conclusion, the only option that is equivalent to the given expression $n^2 + 26n + 88$ is option B, $(n+4)(n+22)$. This is because the coefficients of the linear and constant terms are the same when we multiply the two binomials.

Final Answer

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Introduction

In our previous article, we explored how to find an equivalent expression for a given quadratic expression, specifically $n^2 + 26n + 88$. We examined each option and determined which one is equivalent to the given expression. In this article, we will provide a Q&A guide to help you better understand quadratic expressions and how to manipulate them.

Q: What is a quadratic expression?

A: A quadratic expression is a polynomial of degree two, which means it has a highest power of two. The general form of a quadratic expression is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is equal to the constant term ($c$) and whose sum is equal to the coefficient of the linear term ($b$). You can then express the quadratic expression as a product of two binomials.

Q: What is the difference between factoring and expanding?

A: Factoring involves expressing a quadratic expression as a product of two binomials, while expanding involves multiplying two binomials to get a quadratic expression.

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

A: To determine if two expressions are equivalent, you need to compare their coefficients and constant terms. If the coefficients and constant terms are the same, then the expressions are equivalent.

Q: What is the importance of quadratic expressions in real-life applications?

A: Quadratic expressions have numerous real-life applications, including physics, engineering, economics, and computer science. They are used to model real-world problems, such as the motion of objects, the growth of populations, and the behavior of financial markets.

Q: Can I use quadratic expressions to solve systems of equations?

A: Yes, you can use quadratic expressions to solve systems of equations. By equating the expressions and solving for the variables, you can find the solution to the system.

Q: How do I use quadratic expressions to model real-world problems?

A: To use quadratic expressions to model real-world problems, you need to identify the variables and parameters involved in the problem. You can then use the quadratic expression to represent the relationship between the variables and parameters.

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

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

• Not checking if the expression is a perfect square
• Not factoring the expression correctly
• Not expanding the expression correctly
• Not comparing the coefficients and constant terms correctly

Conclusion

In conclusion, quadratic expressions are a fundamental concept in algebra, and understanding how to manipulate them is crucial for solving various mathematical problems. By following the tips and guidelines provided in this article, you can better understand quadratic expressions and how to use them to model real-world problems.

Final Tips

• Practice, practice, practice: The more you practice working with quadratic expressions, the more comfortable you will become with them.
• Use visual aids: Visual aids such as graphs and charts can help you understand the relationship between the variables and parameters.
• Check your work: Always check your work to ensure that you have factored or expanded the expression correctly.

Additional Resources

• Khan Academy: Quadratic Expressions
• Mathway: Quadratic Expressions
• Wolfram Alpha: Quadratic Expressions

Final Answer

The final answer is that quadratic expressions are a powerful tool for modeling real-world problems and solving mathematical equations. By understanding how to manipulate them, you can better solve problems and make informed decisions.