Which Expression Is Equivalent To $(2x - 6)^2$?

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Introduction

When dealing with algebraic expressions, it's often necessary to simplify or expand them to make them easier to work with. One common operation is expanding a squared binomial, which can be a bit tricky if you're not familiar with the process. In this article, we'll explore how to expand the expression $(2x - 6)^2$ and find an equivalent expression.

Understanding the Problem

The given expression is a squared binomial, which means it's in the form of $(a - b)^2$. To expand this expression, we'll use the formula $(a - b)^2 = a^2 - 2ab + b^2$. In this case, $a = 2x$ and $b = 6$.

Expanding the Expression

To expand the expression, we'll start by squaring the first term, $2x$. This gives us $(2x)^2 = 4x^2$. Next, we'll multiply the first term by the second term, $-6$. This gives us $-12x$. Finally, we'll square the second term, $-6$. This gives us $(-6)^2 = 36$.

Combining the Terms

Now that we have the individual terms, we can combine them to get the expanded expression. We'll start by combining the squared terms, $4x^2$ and $36$. This gives us $4x^2 + 36$. Next, we'll combine the terms with the variable, $-12x$. Since there's only one term with the variable, we can leave it as is.

The Final Expression

After combining the terms, we get the final expression: $4x^2 - 12x + 36$. This is the expanded form of the original expression, $(2x - 6)^2$.

Checking the Answer

To make sure our answer is correct, we can plug it back into the original expression and simplify. We'll start by squaring the binomial: $(2x - 6)^2 = (2x)^2 - 2(2x)(6) + 6^2$. This gives us $4x^2 - 24x + 36$. Since this is equal to our final expression, we know that our answer is correct.

Conclusion

In this article, we explored how to expand the expression $(2x - 6)^2$ and find an equivalent expression. We used the formula $(a - b)^2 = a^2 - 2ab + b^2$ to expand the expression and then combined the terms to get the final expression. We also checked our answer by plugging it back into the original expression and simplifying. With this process, you can expand any squared binomial and find an equivalent expression.

Additional Examples

If you're interested in practicing expanding squared binomials, here are a few additional examples:

• $(3x + 2)^2$
• $(x - 4)^2$
• $(2x + 5)^2$

To expand these expressions, simply follow the same process we used in this article. Square the first term, multiply the first term by the second term, square the second term, and then combine the terms.

Tips and Tricks

When expanding squared binomials, it's often helpful to use the formula $(a - b)^2 = a^2 - 2ab + b^2$. This formula can help you remember the correct order of operations and make the process easier.

Common Mistakes

When expanding squared binomials, it's easy to make mistakes. Here are a few common mistakes to watch out for:

• Forgetting to square the first term
• Forgetting to multiply the first term by the second term
• Forgetting to square the second term
• Not combining the terms correctly

By being aware of these common mistakes, you can avoid them and get the correct answer.

Final Thoughts

Expanding squared binomials can be a bit tricky, but with practice and patience, you can master the process. Remember to use the formula $(a - b)^2 = a^2 - 2ab + b^2$ and to combine the terms correctly. With these tips and tricks, you'll be able to expand any squared binomial and find an equivalent expression.

Introduction

In our previous article, we explored how to expand the expression $(2x - 6)^2$ and find an equivalent expression. We used the formula $(a - b)^2 = a^2 - 2ab + b^2$ to expand the expression and then combined the terms to get the final expression. In this article, we'll answer some common questions about expanding squared binomials.

Q: What is a squared binomial?

A: A squared binomial is an expression in the form of $(a - b)^2$ or $(a + b)^2$. It's a binomial (a polynomial with two terms) that's been squared.

Q: How do I expand a squared binomial?

A: To expand a squared binomial, you can use the formula $(a - b)^2 = a^2 - 2ab + b^2$. Simply square the first term, multiply the first term by the second term, square the second term, and then combine the terms.

Q: What if the binomial is in the form of $(a + b)^2$?

A: If the binomial is in the form of $(a + b)^2$, you can use the formula $(a + b)^2 = a^2 + 2ab + b^2$. Simply square the first term, multiply the first term by the second term, square the second term, and then combine the terms.

Q: How do I know which formula to use?

A: If the binomial is in the form of $(a - b)^2$, use the formula $(a - b)^2 = a^2 - 2ab + b^2$. If the binomial is in the form of $(a + b)^2$, use the formula $(a + b)^2 = a^2 + 2ab + b^2$.

Q: What if I have a squared binomial with variables and constants?

A: If you have a squared binomial with variables and constants, simply expand the expression using the formula and then combine the terms. For example, if you have $(2x + 3)^2$, you can expand it using the formula $(a + b)^2 = a^2 + 2ab + b^2$.

Q: How do I check my answer?

A: To check your answer, plug it back into the original expression and simplify. If the simplified expression is equal to the original expression, then your answer is correct.

Q: What are some common mistakes to watch out for?

A: Some common mistakes to watch out for when expanding squared binomials include:

• Forgetting to square the first term
• Forgetting to multiply the first term by the second term
• Forgetting to square the second term
• Not combining the terms correctly

Q: How can I practice expanding squared binomials?

A: You can practice expanding squared binomials by working through examples and exercises. Try expanding different types of squared binomials, such as $(a - b)^2$ and $(a + b)^2$. You can also use online resources or math textbooks to find additional practice problems.

Q: What are some real-world applications of expanding squared binomials?

A: Expanding squared binomials has many real-world applications, including:

• Algebra: Expanding squared binomials is a fundamental concept in algebra, and it's used to solve equations and inequalities.
• Calculus: Expanding squared binomials is used in calculus to find derivatives and integrals.
• Physics: Expanding squared binomials is used in physics to describe the motion of objects and to calculate forces and energies.
• Engineering: Expanding squared binomials is used in engineering to design and analyze systems, such as bridges and buildings.

Conclusion

Expanding squared binomials is a fundamental concept in algebra, and it has many real-world applications. By understanding how to expand squared binomials, you can solve equations and inequalities, find derivatives and integrals, and describe the motion of objects. With practice and patience, you can master the process of expanding squared binomials and apply it to a wide range of problems.