Factor Completely.$3v^2 - 33v + 72$

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Introduction

Factoring is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two or more polynomials. In this article, we will focus on factoring the quadratic expression $3v^2 - 33v + 72$. Factoring completely means expressing the quadratic expression as a product of linear factors, which can be used to solve equations and inequalities.

Understanding the Quadratic Expression

Before we begin factoring, let's take a closer look at the quadratic expression $3v^2 - 33v + 72$. This expression is a quadratic trinomial, which means it has three terms: a quadratic term, a linear term, and a constant term. The quadratic term is $3v^2$, the linear term is $-33v$, and the constant term is $72$.

Factoring by Grouping

One method of factoring a quadratic expression is by grouping. This method involves grouping the first two terms and the last two terms, and then factoring out a common factor from each group. To factor the quadratic expression $3v^2 - 33v + 72$ by grouping, we can start by factoring out a common factor from the first two terms:

3v^2 - 33v + 72 = (3v^2 - 33v) + 72

Next, we can factor out a common factor from the last two terms:

(3v^2 - 33v) + 72 = 3v(v - 11) + 72

Now, we can see that the expression $3v(v - 11) + 72$ can be factored further by grouping:

3v(v - 11) + 72 = 3v(v - 11) + 9(8)

Finally, we can factor out a common factor from each group:

3v(v - 11) + 9(8) = 3v(v - 11 + 8) = 3v(v - 3)

Factoring Using the Quadratic Formula

Another method of factoring a quadratic expression is by using the quadratic formula. The quadratic formula is a formula that can be used to find the roots of a quadratic equation. To factor the quadratic expression $3v^2 - 33v + 72$ using the quadratic formula, we can start by setting the expression equal to zero:

3v^2 - 33v + 72 = 0

Next, we can use the quadratic formula to find the roots of the equation:

v = (-b ± √(b^2 - 4ac)) / 2a

In this case, $a = 3$, $b = -33$, and $c = 72$. Plugging these values into the quadratic formula, we get:

v = (33 ± √((-33)^2 - 4(3)(72))) / 2(3)

Simplifying the expression, we get:

v = (33 ± √(1089 - 864)) / 6

v = (33 ± √225) / 6

v = (33 ± 15) / 6

Now, we can see that the expression $(33 ± 15) / 6$ can be factored further:

v = (33 + 15) / 6 or v = (33 - 15) / 6

v = 48 / 6 or v = 18 / 6

v = 8 or v = 3

Factoring Completely

Now that we have factored the quadratic expression $3v^2 - 33v + 72$ using both the grouping method and the quadratic formula, we can see that the expression can be factored completely as:

3v^2 - 33v + 72 = 3v(v - 3)(v - 8)

This is the final factored form of the quadratic expression.

Conclusion

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Introduction

In our previous article, we discussed the concept of factoring a quadratic expression and applied it to the quadratic expression $3v^2 - 33v + 72$. 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 means expressing a quadratic expression as a product of linear factors. This involves breaking down the quadratic expression into its simplest form, which can be used to solve equations and inequalities.

Q: Why is factoring completely important?

A: Factoring completely is important because it allows us to solve equations and inequalities more easily. By expressing a quadratic expression as a product of linear factors, we can use the zero product property to find the roots of the equation.

Q: How do I factor a quadratic expression completely?

A: To factor a quadratic expression completely, you can use one of two methods: the grouping method or the quadratic formula. The grouping method involves grouping the first two terms and the last two terms, and then factoring out a common factor from each group. The quadratic formula involves using the quadratic formula to find the roots of the equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to find the roots of a quadratic equation. The formula is:

v = (-b ± √(b^2 - 4ac)) / 2a

Q: How do I use the quadratic formula to factor a quadratic expression?

A: To use the quadratic formula to factor a quadratic expression, you can start by setting the expression equal to zero. Then, you can plug the values of $a$, $b$, and $c$ into the quadratic formula to find the roots of the equation.

Q: What are some common mistakes to avoid when factoring a quadratic expression completely?

A: Some common mistakes to avoid when factoring a quadratic expression completely include:

• Not factoring out a common factor from each group when using the grouping method
• Not using the correct values of $a$, $b$, and $c$ when using the quadratic formula
• Not simplifying the expression after factoring

Q: How do I check my work when factoring a quadratic expression completely?

A: To check your work when factoring a quadratic expression completely, you can multiply the factors together to see if you get the original expression. You can also use the zero product property to check if the factors are correct.

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

A: Some real-world applications of factoring completely include:

• Solving equations and inequalities in physics and engineering
• Modeling population growth and decline in biology
• Analyzing data in statistics and data analysis

Conclusion

In this article, we have provided a Q&A guide to help you understand the concept of factoring completely and how to apply it to different types of quadratic expressions. By following the steps outlined in this article, you can become proficient in factoring completely and apply it to real-world problems.

Additional Resources

For more information on factoring completely, you can check out the following resources:

• Khan Academy: Factoring Quadratic Expressions
• Mathway: Factoring Quadratic Expressions
• Wolfram Alpha: Factoring Quadratic Expressions

Practice Problems

To practice factoring completely, try the following problems:

• Factor the quadratic expression $2x^2 + 5x + 3$ completely.
• Factor the quadratic expression $x^2 - 7x + 12$ completely.
• Factor the quadratic expression $3x^2 + 2x - 5$ completely.

Answer Key

To check your work, you can refer to the following answer key:

• $2x^2 + 5x + 3 = (2x + 3)(x + 1)$
• $x^2 - 7x + 12 = (x - 3)(x - 4)$
• $3x^2 + 2x - 5 = (3x - 5)(x + 1)$