Factor Completely: $\[9g^2 + 30g + 25\\]Express The Answer In The Form \[$(ag + B)^2\$\].Enter Your Answer In The Box: \[$\square\$\]

Introduction

Factoring completely is a fundamental concept in algebra that involves expressing a quadratic expression in the form of a perfect square trinomial. This technique is essential in solving quadratic equations and is widely used in various mathematical applications. In this article, we will focus on factoring completely the quadratic expression $9g^2 + 30g + 25$ and express the answer in the form $(ag + b)^2$.

Understanding the Concept of Factoring Completely

Factoring completely involves expressing a quadratic expression in the form of a perfect square trinomial, which is a quadratic expression that can be written as the square of a binomial. A perfect square trinomial has the form $(ag + b)^2$, where $a$ and $b$ are constants. To factor completely, we need to identify the values of $a$ and $b$ that make the quadratic expression a perfect square trinomial.

Step 1: Identify the Values of $a$ and $b$

To factor completely, we need to identify the values of $a$ and $b$ that make the quadratic expression $9g^2 + 30g + 25$ a perfect square trinomial. We can start by looking at the first term, which is $9g^2$. This term can be written as $(3g)^2$, which is the square of the binomial $3g$. Therefore, we can write the quadratic expression as $(3g)^2 + 30g + 25$.

Step 2: Identify the Middle Term

The middle term of the quadratic expression is $30g$. To make the quadratic expression a perfect square trinomial, we need to find a value of $b$ such that the middle term is equal to $2ab$. In this case, we can see that $2ab = 2(3g)(5) = 30g$. Therefore, we can write the quadratic expression as $(3g)^2 + 2(3g)(5) + 25$.

Step 3: Identify the Last Term

The last term of the quadratic expression is $25$. To make the quadratic expression a perfect square trinomial, we need to find a value of $b$ such that the last term is equal to $b^2$. In this case, we can see that $b^2 = 25$, which means that $b = 5$.

Step 4: Write the Quadratic Expression in the Form $(ag + b)^2$

Now that we have identified the values of $a$ and $b$, we can write the quadratic expression in the form $(ag + b)^2$. In this case, we can write the quadratic expression as $(3g + 5)^2$.

Conclusion

In conclusion, factoring completely involves expressing a quadratic expression in the form of a perfect square trinomial. To factor completely, we need to identify the values of $a$ and $b$ that make the quadratic expression a perfect square trinomial. In this article, we have focused on factoring completely the quadratic expression $9g^2 + 30g + 25$ and expressed the answer in the form $(ag + b)^2$. We have identified the values of $a$ and $b$ and written the quadratic expression in the form $(ag + b)^2$. This technique is essential in solving quadratic equations and is widely used in various mathematical applications.

Example Problems

Example 1

Factor completely the quadratic expression $4x^2 + 12x + 9$.

Solution

To factor completely, we need to identify the values of $a$ and $b$ that make the quadratic expression a perfect square trinomial. We can start by looking at the first term, which is $4x^2$. This term can be written as $(2x)^2$, which is the square of the binomial $2x$. Therefore, we can write the quadratic expression as $(2x)^2 + 12x + 9$.

The middle term of the quadratic expression is $12x$. To make the quadratic expression a perfect square trinomial, we need to find a value of $b$ such that the middle term is equal to $2ab$. In this case, we can see that $2ab = 2(2x)(3) = 12x$. Therefore, we can write the quadratic expression as $(2x)^2 + 2(2x)(3) + 9$.

The last term of the quadratic expression is $9$. To make the quadratic expression a perfect square trinomial, we need to find a value of $b$ such that the last term is equal to $b^2$. In this case, we can see that $b^2 = 9$, which means that $b = 3$.

Now that we have identified the values of $a$ and $b$, we can write the quadratic expression in the form $(ag + b)^2$. In this case, we can write the quadratic expression as $(2x + 3)^2$.

Example 2

Factor completely the quadratic expression $9y^2 + 24y + 16$.

Solution

To factor completely, we need to identify the values of $a$ and $b$ that make the quadratic expression a perfect square trinomial. We can start by looking at the first term, which is $9y^2$. This term can be written as $(3y)^2$, which is the square of the binomial $3y$. Therefore, we can write the quadratic expression as $(3y)^2 + 24y + 16$.

The middle term of the quadratic expression is $24y$. To make the quadratic expression a perfect square trinomial, we need to find a value of $b$ such that the middle term is equal to $2ab$. In this case, we can see that $2ab = 2(3y)(4) = 24y$. Therefore, we can write the quadratic expression as $(3y)^2 + 2(3y)(4) + 16$.

The last term of the quadratic expression is $16$. To make the quadratic expression a perfect square trinomial, we need to find a value of $b$ such that the last term is equal to $b^2$. In this case, we can see that $b^2 = 16$, which means that $b = 4$.

Now that we have identified the values of $a$ and $b$, we can write the quadratic expression in the form $(ag + b)^2$. In this case, we can write the quadratic expression as $(3y + 4)^2$.

Final Answer

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Introduction

Factoring completely is a fundamental concept in algebra that involves expressing a quadratic expression in the form of a perfect square trinomial. In our previous article, we discussed the steps involved in factoring completely and provided examples of how to factor completely the quadratic expression $9g^2 + 30g + 25$. In this article, we will provide a Q&A guide to help you understand the concept of factoring completely and how to apply it to different types of quadratic expressions.

Q: What is factoring completely?

A: Factoring completely involves expressing a quadratic expression in the form of a perfect square trinomial. A perfect square trinomial is a quadratic expression that can be written as the square of a binomial.

Q: How do I know if a quadratic expression can be factored completely?

A: To determine if a quadratic expression can be factored completely, you need to check if the quadratic expression can be written as the square of a binomial. You can do this by looking at the first term of the quadratic expression and checking if it can be written as the square of a binomial.

Q: What are the steps involved in factoring completely?

A: The steps involved in factoring completely are:

1. Identify the values of $a$ and $b$ that make the quadratic expression a perfect square trinomial.
2. Write the quadratic expression in the form $(ag + b)^2$.

Q: How do I identify the values of $a$ and $b$?

A: To identify the values of $a$ and $b$, you need to look at the first term of the quadratic expression and check if it can be written as the square of a binomial. You can also use the formula $a^2 = \frac{b^2 - c^2}{4ac}$ to find the value of $a$.

Q: What is the formula for factoring completely?

A: The formula for factoring completely is:

$(ag + b)^2 = a^2g^2 + 2abg + b^2$

Q: Can I factor completely a quadratic expression with a negative leading coefficient?

A: Yes, you can factor completely a quadratic expression with a negative leading coefficient. To do this, you need to follow the same steps as before, but you need to take the negative sign into account when writing the quadratic expression in the form $(ag + b)^2$.

Q: Can I factor completely a quadratic expression with a coefficient of 1?

A: Yes, you can factor completely a quadratic expression with a coefficient of 1. To do this, you need to follow the same steps as before, but you need to take the coefficient of 1 into account when writing the quadratic expression in the form $(ag + b)^2$.

Q: Can I factor completely a quadratic expression with a coefficient of -1?

A: Yes, you can factor completely a quadratic expression with a coefficient of -1. To do this, you need to follow the same steps as before, but you need to take the negative sign into account when writing the quadratic expression in the form $(ag + b)^2$.

Q: Can I factor completely a quadratic expression with a coefficient of 0?

A: No, you cannot factor completely a quadratic expression with a coefficient of 0. This is because a quadratic expression with a coefficient of 0 is not a quadratic expression.

Q: Can I factor completely a quadratic expression with a coefficient of 1 and a negative leading coefficient?

A: Yes, you can factor completely a quadratic expression with a coefficient of 1 and a negative leading coefficient. To do this, you need to follow the same steps as before, but you need to take the negative sign into account when writing the quadratic expression in the form $(ag + b)^2$.

Q: Can I factor completely a quadratic expression with a coefficient of -1 and a negative leading coefficient?

A: Yes, you can factor completely a quadratic expression with a coefficient of -1 and a negative leading coefficient. To do this, you need to follow the same steps as before, but you need to take the negative sign into account when writing the quadratic expression in the form $(ag + b)^2$.

Q: Can I factor completely a quadratic expression with a coefficient of 0 and a negative leading coefficient?

A: No, you cannot factor completely a quadratic expression with a coefficient of 0 and a negative leading coefficient. This is because a quadratic expression with a coefficient of 0 is not a quadratic expression.

Conclusion

In conclusion, factoring completely is a fundamental concept in algebra that involves expressing a quadratic expression in the form of a perfect square trinomial. By following the steps outlined in this article, you can factor completely a quadratic expression and express it in the form $(ag + b)^2$. We hope that this Q&A guide has helped you understand the concept of factoring completely and how to apply it to different types of quadratic expressions.

Final Answer

The final answer is: $\boxed{(3g + 5)^2}$