Factor $m^2 - 49$.

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Introduction

In mathematics, factoring is a fundamental concept that involves expressing an algebraic expression as a product of simpler expressions. In this article, we will focus on factoring the expression $m^2 - 49$. This expression is a difference of squares, which is a common algebraic identity that can be factored using a specific formula.

Understanding the Difference of Squares

The difference of squares is a fundamental algebraic identity that states:

$a^2 - b^2 = (a + b)(a - b)$

This identity can be used to factor any expression of the form $a^2 - b^2$, where $a$ and $b$ are any real numbers.

Factoring $m^2 - 49$

To factor the expression $m^2 - 49$, we can use the difference of squares identity. In this case, we have:

$m^2 - 49 = (m + 7)(m - 7)$

This is because $49$ is the square of $7$, and we can rewrite the expression as:

$m^2 - 7^2 = (m + 7)(m - 7)$

Verifying the Factorization

To verify that the factorization is correct, we can multiply the two factors together:

$(m + 7)(m - 7) = m^2 - 7m + 7m - 49$

Simplifying the expression, we get:

$m^2 - 49$

This shows that the factorization is correct, and we have successfully factored the expression $m^2 - 49$.

Real-World Applications

Factoring expressions like $m^2 - 49$ has many real-world applications in mathematics and science. For example, in algebra, factoring is used to solve equations and inequalities. In calculus, factoring is used to find the derivative of a function. In physics, factoring is used to solve problems involving motion and energy.

Conclusion

In conclusion, factoring the expression $m^2 - 49$ is a simple process that involves using the difference of squares identity. By understanding this identity and applying it to the expression, we can factor it into the product of two simpler expressions. This is a fundamental concept in mathematics that has many real-world applications.

Additional Examples

Here are a few additional examples of factoring expressions using the difference of squares identity:

• $x^2 - 16 = (x + 4)(x - 4)$
• $y^2 - 9 = (y + 3)(y - 3)$
• $z^2 - 25 = (z + 5)(z - 5)$

These examples demonstrate how the difference of squares identity can be used to factor a wide range of expressions.

Tips and Tricks

Here are a few tips and tricks for factoring expressions using the difference of squares identity:

• Make sure to identify the square root of the constant term in the expression.
• Use the difference of squares identity to factor the expression.
• Verify the factorization by multiplying the two factors together.

By following these tips and tricks, you can become proficient in factoring expressions using the difference of squares identity.

Common Mistakes

Here are a few common mistakes to avoid when factoring expressions using the difference of squares identity:

• Failing to identify the square root of the constant term in the expression.
• Not using the difference of squares identity to factor the expression.
• Not verifying the factorization by multiplying the two factors together.

By avoiding these common mistakes, you can ensure that your factorizations are correct and accurate.

Conclusion

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Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about factoring the expression $m^2 - 49$. Whether you are a student, teacher, or simply someone who wants to learn more about mathematics, this article is for you.

Q: What is the difference of squares identity?

A: The difference of squares identity is a fundamental algebraic identity that states:

$a^2 - b^2 = (a + b)(a - b)$

This identity can be used to factor any expression of the form $a^2 - b^2$, where $a$ and $b$ are any real numbers.

Q: How do I factor $m^2 - 49$ using the difference of squares identity?

A: To factor $m^2 - 49$ using the difference of squares identity, you need to identify the square root of the constant term in the expression. In this case, the constant term is $49$, which is the square of $7$. Therefore, you can rewrite the expression as:

$m^2 - 7^2 = (m + 7)(m - 7)$

Q: Why is it important to verify the factorization?

A: Verifying the factorization is important because it ensures that the factorization is correct. If you multiply the two factors together, you should get the original expression. In this case, multiplying $(m + 7)$ and $(m - 7)$ together gives:

$(m + 7)(m - 7) = m^2 - 7m + 7m - 49$

Simplifying the expression, you get:

$m^2 - 49$

This shows that the factorization is correct.

Q: What are some real-world applications of factoring expressions like $m^2 - 49$?

A: Factoring expressions like $m^2 - 49$ has many real-world applications in mathematics and science. For example, in algebra, factoring is used to solve equations and inequalities. In calculus, factoring is used to find the derivative of a function. In physics, factoring is used to solve problems involving motion and energy.

Q: What are some common mistakes to avoid when factoring expressions like $m^2 - 49$?

A: Some common mistakes to avoid when factoring expressions like $m^2 - 49$ include:

• Failing to identify the square root of the constant term in the expression.
• Not using the difference of squares identity to factor the expression.
• Not verifying the factorization by multiplying the two factors together.

Q: How can I practice factoring expressions like $m^2 - 49$?

A: There are many ways to practice factoring expressions like $m^2 - 49$. Here are a few suggestions:

• Practice factoring expressions with different constant terms, such as $x^2 - 16$ or $y^2 - 9$.
• Use online resources, such as math websites or apps, to practice factoring expressions.
• Work with a partner or tutor to practice factoring expressions together.

Q: What are some additional tips and tricks for factoring expressions like $m^2 - 49$?

A: Here are a few additional tips and tricks for factoring expressions like $m^2 - 49$:

• Make sure to identify the square root of the constant term in the expression.
• Use the difference of squares identity to factor the expression.
• Verify the factorization by multiplying the two factors together.
• Practice factoring expressions with different constant terms.
• Use online resources to practice factoring expressions.

Conclusion

In conclusion, factoring the expression $m^2 - 49$ is a simple process that involves using the difference of squares identity. By understanding this identity and applying it to the expression, we can factor it into the product of two simpler expressions. This is a fundamental concept in mathematics that has many real-world applications.