Factor Completely: $16a^2 + 48a + 36$A. $16(2a-3)(2a+3$\] B. $4(2a+3)^2$ C. $4(a-1)^2$ D. $4(3a+1)^2$

Understanding Quadratic Expressions

A quadratic expression is a polynomial expression of degree two, which means the highest power of the variable is two. It is typically written in the form of $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. In this article, we will focus on factoring a quadratic expression of the form $16a^2 + 48a + 36$.

The Process of Factoring

Factoring a quadratic expression involves expressing it as a product of two or more expressions. This can be done by finding the greatest common factor (GCF) of the terms, or by using the method of grouping. In this case, we will use the method of grouping to factor the given quadratic expression.

Step 1: Factor Out the Greatest Common Factor

The first step in factoring the quadratic expression is to factor out the greatest common factor (GCF) of the terms. In this case, the GCF of $16a^2$, $48a$, and $36$ is $4$. Factoring out the GCF, we get:

$$16a^2 + 48a + 36 = 4(4a^2 + 12a + 9)$$

Step 2: Factor the Quadratic Expression Inside the Parentheses

Now that we have factored out the GCF, we can focus on factoring the quadratic expression inside the parentheses. To do this, we need to find two numbers whose product is $4 \times 9 = 36$ and whose sum is $12$. These numbers are $6$ and $6$, so we can write the quadratic expression as:

$$4a^2 + 12a + 9 = (2a + 3)^2$$

Step 3: Write the Final Factored Form

Now that we have factored the quadratic expression inside the parentheses, we can write the final factored form of the original expression:

$$16a^2 + 48a + 36 = 4(2a + 3)^2$$

Conclusion

In this article, we have factored the quadratic expression $16a^2 + 48a + 36$ using the method of grouping. We first factored out the greatest common factor (GCF) of the terms, and then factored the quadratic expression inside the parentheses. The final factored form of the expression is $4(2a + 3)^2$.

Answer

The correct answer is B. $4(2a+3)^2$.

Comparison of Options

Let's compare the options given in the problem:

A. $16(2a-3)(2a+3)$ B. $4(2a+3)^2$ C. $4(a-1)^2$ D. $4(3a+1)^2$

Option A is incorrect because the expression inside the parentheses is not a perfect square.

Option C is incorrect because the expression inside the parentheses is not a perfect square.

Option D is incorrect because the expression inside the parentheses is not a perfect square.

Option B is the correct answer because it is the final factored form of the original expression.

Final Thoughts

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Understanding Quadratic Expressions

A quadratic expression is a polynomial expression of degree two, which means the highest power of the variable is two. It is typically written in the form of $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. In this article, we will focus on factoring quadratic expressions.

Q: What is the process of factoring a quadratic expression?

A: The process of factoring a quadratic expression involves expressing it as a product of two or more expressions. This can be done by finding the greatest common factor (GCF) of the terms, or by using the method of grouping.

Q: How do I find the greatest common factor (GCF) of the terms?

A: To find the GCF of the terms, you need to identify the largest factor that divides all the terms evenly. For example, if you have the terms $16a^2$, $48a$, and $36$, the GCF is $4$.

Q: What is the method of grouping?

A: The method of grouping involves factoring the quadratic expression by grouping the terms in pairs. For example, if you have the expression $16a^2 + 48a + 36$, you can group the terms as $(16a^2 + 36a) + (12a + 36)$.

Q: How do I factor the quadratic expression inside the parentheses?

A: To factor the quadratic expression inside the parentheses, you need to find two numbers whose product is the product of the coefficients of the terms and whose sum is the coefficient of the middle term. For example, if you have the expression $(16a^2 + 36a)$, you can factor it as $(8a(2a + 3))$.

Q: What is a perfect square trinomial?

A: A perfect square trinomial is a quadratic expression that can be factored as the square of a binomial. For example, the expression $(2a + 3)^2$ is a perfect square trinomial.

Q: How do I identify a perfect square trinomial?

A: To identify a perfect square trinomial, you need to check if the expression can be factored as the square of a binomial. You can do this by checking if the product of the coefficients of the terms is a perfect square and if the sum of the coefficients of the middle term is twice the square root of the product of the coefficients.

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

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

• Not finding the greatest common factor (GCF) of the terms
• Not grouping the terms correctly
• Not identifying perfect square trinomials
• Not checking if the expression can be factored as the square of a binomial

Q: How do I check if a quadratic expression can be factored as the square of a binomial?

A: To check if a quadratic expression can be factored as the square of a binomial, you need to check if the product of the coefficients of the terms is a perfect square and if the sum of the coefficients of the middle term is twice the square root of the product of the coefficients.

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

A: Some real-world applications of factoring quadratic expressions include:

• Solving quadratic equations
• Finding the maximum or minimum value of a quadratic function
• Modeling real-world situations using quadratic functions

Conclusion

In this article, we have discussed the process of factoring quadratic expressions and provided answers to some common questions. We have also highlighted some common mistakes to avoid and provided some real-world applications of factoring quadratic expressions.