What Is The Factorization Of $49b^2 - 81$?A. $(7b - 9)(7b - 9)$ B. \$(7b - 9)(7b + 9)$[/tex\] C. $\left(7b^2 - 9\right)\left(7b^2 - 9\right)$ D. $\left(7b^2 - 9\right)\left(7b^2 + 9\right)$

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Understanding the Problem

The problem requires us to find the factorization of the expression 49b2−8149b^2 - 81. This involves breaking down the given expression into simpler components, known as factors, that when multiplied together, result in the original expression.

Identifying the Type of Expression

The given expression 49b2−8149b^2 - 81 is a quadratic expression, as it consists of a squared variable (b2b^2) and a constant term (−81-81). This type of expression can often be factored using various techniques, such as the difference of squares or the sum/difference of cubes.

Applying the Difference of Squares Formula

One of the most common techniques for factoring quadratic expressions is the difference of squares formula, which states that:

a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)

In this case, we can rewrite the given expression as:

49b2−81=(7b)2−9249b^2 - 81 = (7b)^2 - 9^2

Now, we can apply the difference of squares formula to factor the expression:

(7b)2−92=(7b+9)(7b−9)(7b)^2 - 9^2 = (7b + 9)(7b - 9)

Evaluating the Answer Choices

Now that we have factored the expression, we can compare our result with the answer choices provided:

A. $(7b - 9)(7b - 9)$ B. $(7b - 9)(7b + 9)$ C. $(7b^2 - 9)(7b^2 - 9)$ D. $(7b^2 - 9)(7b^2 + 9)$

Our factored expression matches answer choice B, which is:

(7b−9)(7b+9)(7b - 9)(7b + 9)

Conclusion

In conclusion, the factorization of the expression 49b2−8149b^2 - 81 is indeed $(7b - 9)(7b + 9)$. This result was obtained by applying the difference of squares formula to the given expression, which is a common technique for factoring quadratic expressions.

Key Takeaways

  • The difference of squares formula is a useful technique for factoring quadratic expressions.
  • The formula states that a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b).
  • This formula can be applied to expressions of the form (7b)2−92(7b)^2 - 9^2 to obtain the factored form (7b+9)(7b−9)(7b + 9)(7b - 9).

Real-World Applications

Factoring quadratic expressions is an essential skill in mathematics, with numerous real-world applications in fields such as physics, engineering, and economics. For example, factoring quadratic expressions can be used to solve problems involving projectile motion, electrical circuits, and financial modeling.

Common Mistakes to Avoid

When factoring quadratic expressions, it's essential to avoid common mistakes such as:

  • Not recognizing the difference of squares pattern
  • Applying the formula incorrectly
  • Failing to check the answer choices

By being aware of these potential pitfalls, you can ensure that your factoring skills are accurate and reliable.

Additional Practice Problems

If you're looking for more practice problems to reinforce your understanding of factoring quadratic expressions, consider trying the following:

  • Factor the expression 16x2−2516x^2 - 25
  • Factor the expression 9y2−49y^2 - 4
  • Factor the expression 25z2−3625z^2 - 36

By practicing these problems, you'll become more confident in your ability to factor quadratic expressions and apply the difference of squares formula with ease.

Q: What is the difference of squares formula?

A: The difference of squares formula is a mathematical formula that states:

a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)

This formula can be used to factor quadratic expressions of the form a2−b2a^2 - b^2.

Q: How do I apply the difference of squares formula?

A: To apply the difference of squares formula, you need to identify the values of aa and bb in the given expression. Then, you can rewrite the expression in the form a2−b2a^2 - b^2 and apply the formula to factor it.

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

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

  • Not recognizing the difference of squares pattern
  • Applying the formula incorrectly
  • Failing to check the answer choices

Q: Can I use the difference of squares formula to factor expressions that are not in the form a2−b2a^2 - b^2?

A: No, the difference of squares formula can only be used to factor expressions that are in the form a2−b2a^2 - b^2. If the expression is not in this form, you may need to use a different factoring technique.

Q: How do I determine if an expression is a perfect square trinomial?

A: To determine if an expression is a perfect square trinomial, you need to check if it can be written in the form (a+b)2(a + b)^2 or (a−b)2(a - b)^2. If it can be written in one of these forms, then it is a perfect square trinomial.

Q: What is the difference between a perfect square trinomial and a quadratic expression?

A: A perfect square trinomial is a quadratic expression that can be written in the form (a+b)2(a + b)^2 or (a−b)2(a - b)^2. A quadratic expression is a polynomial of degree 2, but it may not be a perfect square trinomial.

Q: Can I use the difference of squares formula to factor expressions that have a coefficient of 1?

A: Yes, you can use the difference of squares formula to factor expressions that have a coefficient of 1. For example, the expression x2−4x^2 - 4 can be factored using the difference of squares formula as (x+2)(x−2)(x + 2)(x - 2).

Q: How do I factor expressions that have a coefficient other than 1?

A: To factor expressions that have a coefficient other than 1, you need to factor out the coefficient first. For example, the expression 2x2−82x^2 - 8 can be factored by first factoring out the coefficient 2, resulting in 2(x2−4)2(x^2 - 4).

Q: Can I use the difference of squares formula to factor expressions that have a variable in the denominator?

A: No, you cannot use the difference of squares formula to factor expressions that have a variable in the denominator. This is because the formula requires the expression to be in the form a2−b2a^2 - b^2, and expressions with variables in the denominator are not in this form.

Q: How do I factor expressions that have a variable in the denominator?

A: To factor expressions that have a variable in the denominator, you need to use a different factoring technique, such as factoring by grouping or using the rational root theorem.

Q: Can I use the difference of squares formula to factor expressions that have a negative sign in front of the expression?

A: Yes, you can use the difference of squares formula to factor expressions that have a negative sign in front of the expression. For example, the expression −x2+4-x^2 + 4 can be factored using the difference of squares formula as −(x+2)(x−2)-(x + 2)(x - 2).

Q: How do I factor expressions that have a negative sign in front of the expression?

A: To factor expressions that have a negative sign in front of the expression, you need to factor the expression inside the parentheses first, and then apply the difference of squares formula.