Factor The Expression:${ N^2 + 16n + 63 }$A. { (n+7)(n+9)$}$B. { (n-7)(n+4)$}$C. { (n-3)(n-10)$}$D. { (n+7)(n-9)$}$

Introduction

Factoring quadratic expressions is a fundamental concept in algebra that allows us to simplify complex expressions and solve equations. In this article, we will focus on factoring the expression $n^2 + 16n + 63$ and explore the different methods and techniques used to factor quadratic expressions.

What is Factoring?

Factoring is the process of expressing a quadratic expression as a product of two or more binomial expressions. This is achieved by finding the factors of the quadratic expression that, when multiplied together, result in the original expression. Factoring quadratic expressions is an essential skill in algebra, as it allows us to simplify complex expressions, solve equations, and graph functions.

The Expression to be Factored

The expression we will be factoring is $n^2 + 16n + 63$. This is a quadratic expression in the form of $ax^2 + bx + c$, where $a = 1$, $b = 16$, and $c = 63$.

Method 1: Factoring by Grouping

One method of factoring quadratic expressions is by grouping. This involves grouping the terms of the expression into two pairs and then factoring out the common factors from each pair.

To factor the expression $n^2 + 16n + 63$ by grouping, we can start by grouping the first two terms, $n^2$ and $16n$, and then factoring out the common factor from each pair.

$n^2 + 16n + 63$

Group the first two terms:

$n^2 + 16n$

Factor out the common factor from each pair:

$n(n + 16) + 63$

Now, we can see that the expression can be factored as:

$n(n + 16) + 63$

However, this is not one of the answer choices. We need to try another method.

Method 2: Factoring by Finding Two Numbers

Another method of factoring quadratic expressions is by finding two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

In this case, we need to find two numbers whose product is equal to $63$ and whose sum is equal to $16$.

The two numbers that satisfy this condition are $7$ and $9$, since $7 \times 9 = 63$ and $7 + 9 = 16$.

Therefore, we can write the expression as:

$n^2 + 16n + 63 = (n + 7)(n + 9)$

This is one of the answer choices.

Conclusion

In this article, we have explored the different methods and techniques used to factor quadratic expressions. We have factored the expression $n^2 + 16n + 63$ using two different methods: factoring by grouping and factoring by finding two numbers. We have also seen that the correct factorization of the expression is $(n + 7)(n + 9)$.

Answer

The correct answer is:

A. $(n+7)(n+9)$

Additional Tips and Tricks

• When factoring quadratic expressions, it is essential to look for two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.
• Factoring by grouping can be a useful method for factoring quadratic expressions, but it may not always result in the correct factorization.
• To factor a quadratic expression, you can try using the quadratic formula or completing the square, but these methods may be more complex and time-consuming.

Common Mistakes to Avoid

• When factoring quadratic expressions, it is essential to avoid making mistakes such as factoring out the wrong term or forgetting to include a term.
• It is also essential to check your work by multiplying the factors together to ensure that you get the original expression.
• If you are having trouble factoring a quadratic expression, try using a different method or seeking help from a teacher or tutor.

Real-World Applications

Factoring quadratic expressions has numerous real-world applications, including:

• Solving equations and inequalities
• Graphing functions
• Finding the maximum or minimum value of a function
• Modeling real-world situations, such as the motion of an object or the growth of a population.

Conclusion

Introduction

Factoring quadratic expressions is a fundamental concept in algebra that allows us to simplify complex expressions and solve equations. In this article, we will provide a Q&A guide to help you understand the different methods and techniques used to factor quadratic expressions.

Q: What is factoring?

A: Factoring is the process of expressing a quadratic expression as a product of two or more binomial expressions. This is achieved by finding the factors of the quadratic expression that, when multiplied together, result in the original expression.

Q: What are the different methods of factoring quadratic expressions?

A: There are several methods of factoring quadratic expressions, including:

• Factoring by grouping
• Factoring by finding two numbers
• Factoring using the quadratic formula
• Factoring using completing the square

Q: How do I factor a quadratic expression using the factoring by grouping method?

A: To factor a quadratic expression using the factoring by grouping method, you can start by grouping the first two terms of the expression and then factoring out the common factor from each pair.

For example, consider the expression $n^2 + 16n + 63$. To factor this expression using the factoring by grouping method, you can start by grouping the first two terms:

$n^2 + 16n$

Then, you can factor out the common factor from each pair:

$n(n + 16) + 63$

However, this is not one of the answer choices. You need to try another method.

Q: How do I factor a quadratic expression using the factoring by finding two numbers method?

A: To factor a quadratic expression using the factoring by finding two numbers method, you need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

For example, consider the expression $n^2 + 16n + 63$. To factor this expression using the factoring by finding two numbers method, you need to find two numbers whose product is equal to $63$ and whose sum is equal to $16$.

The two numbers that satisfy this condition are $7$ and $9$, since $7 \times 9 = 63$ and $7 + 9 = 16$.

Therefore, you can write the expression as:

$n^2 + 16n + 63 = (n + 7)(n + 9)$

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

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

• Factoring out the wrong term
• Forgetting to include a term
• Not checking your work by multiplying the factors together
• Not using the correct method for the given expression

Q: How do I know which method to use when factoring a quadratic expression?

A: The method you use to factor a quadratic expression will depend on the specific expression you are working with. Here are some general guidelines to help you choose the correct method:

• If the expression can be easily grouped, use the factoring by grouping method.
• If the expression can be easily factored using two numbers, use the factoring by finding two numbers method.
• If the expression is a perfect square trinomial, use the factoring using the quadratic formula method.
• If the expression is not a perfect square trinomial, use the factoring using completing the square method.

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

A: Factoring quadratic expressions has numerous real-world applications, including:

• Solving equations and inequalities
• Graphing functions
• Finding the maximum or minimum value of a function
• Modeling real-world situations, such as the motion of an object or the growth of a population.

Conclusion

In conclusion, factoring quadratic expressions is a fundamental concept in algebra that allows us to simplify complex expressions and solve equations. By understanding the different methods and techniques used to factor quadratic expressions, we can become more proficient in solving equations and inequalities, graphing functions, and modeling real-world situations.