Select The Expression That Is Equivalent To \[$(x+6)^2\$\].A. \[$x^2 + 12x^2 + 36\$\] B. \[$x^2 + 12x + 36\$\] C. \[$2x + 12x + 12\$\] D. \[$x^2 + 6x + 36\$\]

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Introduction

In algebra, expanding quadratic expressions is a crucial skill that helps us simplify complex equations and solve problems efficiently. In this article, we will focus on expanding the expression {(x+6)^2$}$ and selecting the equivalent expression from the given options.

Understanding the Concept of Expanding Quadratic Expressions

Expanding quadratic expressions involves multiplying the terms inside the parentheses by each other and then combining like terms. This process helps us simplify the expression and make it easier to work with.

Expanding the Expression {(x+6)^2$}$

To expand the expression {(x+6)^2$}$, we need to multiply the terms inside the parentheses by each other. We can do this using the distributive property, which states that {a(b+c) = ab + ac$}$.

Using the distributive property, we can expand the expression as follows:

{(x+6)^2 = (x+6)(x+6)$}$

{= x(x+6) + 6(x+6)$}$

{= x^2 + 6x + 6x + 36$}$

{= x^2 + 12x + 36$}$

Analyzing the Options

Now that we have expanded the expression {(x+6)^2$}$, let's analyze the options given:

A. {x^2 + 12x^2 + 36$}$

B. {x^2 + 12x + 36$}$

C. ${2x + 12x + 12\$}

D. {x^2 + 6x + 36$}$

Selecting the Equivalent Expression

Based on our expansion of the expression {(x+6)^2$}$, we can see that the correct equivalent expression is:

B. {x^2 + 12x + 36$}$

This is because the expanded expression {x^2 + 12x + 36$}$ matches the expression we obtained by multiplying the terms inside the parentheses.

Conclusion

In conclusion, expanding quadratic expressions is an essential skill in algebra that helps us simplify complex equations and solve problems efficiently. By understanding the concept of expanding quadratic expressions and applying the distributive property, we can expand expressions like {(x+6)^2$}$ and select the equivalent expression from the given options.

Common Mistakes to Avoid

When expanding quadratic expressions, it's essential to avoid common mistakes like:

  • Forgetting to multiply the terms inside the parentheses by each other
  • Not combining like terms correctly
  • Not using the distributive property correctly

By avoiding these common mistakes, we can ensure that our expanded expressions are accurate and help us solve problems efficiently.

Real-World Applications

Expanding quadratic expressions has numerous real-world applications in fields like physics, engineering, and economics. For example, in physics, expanding quadratic expressions can help us model the motion of objects under the influence of gravity or other forces. In engineering, expanding quadratic expressions can help us design and optimize systems like bridges, buildings, and electronic circuits.

Final Thoughts

In conclusion, expanding quadratic expressions is a crucial skill in algebra that helps us simplify complex equations and solve problems efficiently. By understanding the concept of expanding quadratic expressions and applying the distributive property, we can expand expressions like {(x+6)^2$}$ and select the equivalent expression from the given options. Remember to avoid common mistakes and apply the distributive property correctly to ensure accurate results.

Additional Resources

For further practice and review, we recommend the following resources:

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

Introduction

In our previous article, we discussed expanding quadratic expressions and selecting the equivalent expression from the given options. In this article, we will provide a Q&A guide to help you better understand the concept of expanding quadratic expressions and apply it to real-world problems.

Q: What is a quadratic expression?

A: A quadratic expression is a polynomial expression of degree two, which means it has a squared variable (x^2) and possibly linear terms (x) and constant terms.

Q: How do I expand a quadratic expression?

A: To expand a quadratic expression, you need to multiply the terms inside the parentheses by each other and then combine like terms. This process is called the distributive property.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that a(b+c) = ab + ac. This means that you can multiply a single term by each term inside the parentheses separately.

Q: How do I apply the distributive property to expand a quadratic expression?

A: To apply the distributive property, you need to multiply the terms inside the parentheses by each other. For example, to expand the expression (x+6)^2, you would multiply x by x and 6 by x, and then combine the like terms.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, 2x and 3x 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, to combine the terms 2x and 3x, you would add the coefficients 2 and 3 to get 5x.

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

A: Some common mistakes to avoid when expanding quadratic expressions include:

  • Forgetting to multiply the terms inside the parentheses by each other
  • Not combining like terms correctly
  • Not using the distributive property correctly

Q: How do I check my work when expanding quadratic expressions?

A: To check your work, you can plug in a value for the variable and see if the expression simplifies to the correct value. You can also use a calculator or a computer program to check your work.

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

A: Expanding quadratic expressions has numerous real-world applications in fields like physics, engineering, and economics. For example, in physics, expanding quadratic expressions can help us model the motion of objects under the influence of gravity or other forces. In engineering, expanding quadratic expressions can help us design and optimize systems like bridges, buildings, and electronic circuits.

Q: How can I practice expanding quadratic expressions?

A: You can practice expanding quadratic expressions by working through exercises and problems in a textbook or online resource. You can also try creating your own problems and solving them to practice your skills.

Conclusion

In conclusion, expanding quadratic expressions is a crucial skill in algebra that helps us simplify complex equations and solve problems efficiently. By understanding the concept of expanding quadratic expressions and applying the distributive property, we can expand expressions like (x+6)^2 and select the equivalent expression from the given options. Remember to avoid common mistakes and apply the distributive property correctly to ensure accurate results.

Additional Resources

For further practice and review, we recommend the following resources:

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

By practicing and reviewing these resources, you can improve your skills in expanding quadratic expressions and apply them to real-world problems.