Factor Completely.$\[ X^2 - 2x - 24 \\]Enter Your Answer In The Box. $\[\square\\]

Feb 26, 2025 by ADMIN 84 views

Factoring a Quadratic Expression: A Step-by-Step Guide

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Introduction

Factoring a quadratic expression is a fundamental concept in algebra that involves expressing a quadratic equation in the form of a product of two binomials. This technique is essential in solving quadratic equations, simplifying expressions, and understanding the properties of quadratic functions. In this article, we will focus on factoring the quadratic expression $x^2 - 2x - 24$ .

What is Factoring?

Factoring is the process of expressing a quadratic expression as a product of two binomials. A quadratic expression is a polynomial of degree two, which means it has a highest power of $x$ equal to two. The general form of a quadratic expression is $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.

The Factorization Process

To factor a quadratic expression, we need to find two binomials whose product is equal to the original expression. The process involves finding the factors of the constant term, the coefficient of the linear term, and the coefficient of the quadratic term. We will use the following steps to factor the quadratic expression $x^2 - 2x - 24$ :

Step 1: Find the Factors of the Constant Term

The constant term in the quadratic expression is $-24$ . We need to find two numbers whose product is equal to $-24$ and whose sum is equal to the coefficient of the linear term, which is $-2$ .

Step 2: Find the Factors of the Coefficient of the Linear Term

The coefficient of the linear term is $-2$ . We need to find two numbers whose product is equal to the constant term, $-24$ , and whose sum is equal to the coefficient of the linear term, $-2$ .

Step 3: Write the Factored Form

Once we have found the factors of the constant term and the coefficient of the linear term, we can write the factored form of the quadratic expression.

Factoring the Quadratic Expression

Let's apply the steps above to factor the quadratic expression $x^2 - 2x - 24$ .

Step 1: Find the Factors of the Constant Term

The constant term in the quadratic expression is $-24$ . We need to find two numbers whose product is equal to $-24$ and whose sum is equal to the coefficient of the linear term, which is $-2$ .

The factors of $-24$ are: $1, -24$ , $2, -12$ , $3, -8$ , $4, -6$ , $-1, 24$ , $-2, 12$ , $-3, 8$ , and $-4, 6$ .

Step 2: Find the Factors of the Coefficient of the Linear Term

The coefficient of the linear term is $-2$ . We need to find two numbers whose product is equal to the constant term, $-24$ , and whose sum is equal to the coefficient of the linear term, $-2$ .

The factors of $-24$ that add up to $-2$ are: $-6$ and $4$ .

Step 3: Write the Factored Form

Now that we have found the factors of the constant term and the coefficient of the linear term, we can write the factored form of the quadratic expression.

The factored form of the quadratic expression $x^2 - 2x - 24$ is:

\boxed{(x - 6)(x + 4)}

Conclusion

Factoring a quadratic expression is a powerful technique that involves expressing a quadratic equation in the form of a product of two binomials. By following the steps outlined above, we can factor the quadratic expression $x^2 - 2x - 24$ and write it in its factored form. This technique is essential in solving quadratic equations, simplifying expressions, and understanding the properties of quadratic functions.

Frequently Asked Questions

Q: What is factoring?

A: Factoring is the process of expressing a quadratic expression as a product of two binomials.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two binomials whose product is equal to the original expression. You can use the following steps:

Find the factors of the constant term.
Find the factors of the coefficient of the linear term.
Write the factored form.

Q: What are the factors of a quadratic expression?

A: The factors of a quadratic expression are the two binomials whose product is equal to the original expression.

Q: How do I find the factors of a quadratic expression?

A: To find the factors of a quadratic expression, 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.

Example Problems

Problem 1: Factor the quadratic expression $x^2 + 5x + 6$

To factor the quadratic expression $x^2 + 5x + 6$ , we need to find two numbers whose product is equal to $6$ and whose sum is equal to $5$ .

The factors of $6$ are: $1, 6$ , $2, 3$ , $-1, -6$ , $-2, -3$ , $-6, -1$ , $-3, -2$ , and $-6, -1$ .

The factors of $6$ that add up to $5$ are: $2$ and $3$ .

The factored form of the quadratic expression $x^2 + 5x + 6$ is:

\boxed{(x + 2)(x + 3)}

Problem 2: Factor the quadratic expression $x^2 - 7x - 18$

To factor the quadratic expression $x^2 - 7x - 18$ , we need to find two numbers whose product is equal to $-18$ and whose sum is equal to $-7$ .

The factors of $-18$ are: $1, -18$ , $2, -9$ , $3, -6$ , $-1, 18$ , $-2, 9$ , $-3, 6$ , $-6, 3$ , $-9, 2$ , and $-18, 1$ .

The factors of $-18$ that add up to $-7$ are: $-9$ and $2$ .

The factored form of the quadratic expression $x^2 - 7x - 18$ is:

\boxed{(x - 9)(x + 2)}

Practice Problems

Problem 1: Factor the quadratic expression $x^2 + 3x - 4$

Problem 2: Factor the quadratic expression $x^2 - 2x - 15$

Problem 3: Factor the quadratic expression $x^2 + 5x + 4$

Problem 4: Factor the quadratic expression $x^2 - 7x - 12$

Problem 5: Factor the quadratic expression $x^2 + 2x - 15$

Conclusion

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Introduction

Factoring quadratic expressions is a fundamental concept in algebra that involves expressing a quadratic equation in the form of a product of two binomials. In our previous article, we discussed the steps involved in factoring a quadratic expression and provided examples of how to factor different types of quadratic expressions. In this article, we will provide a Q&A guide to help you understand the concept of factoring quadratic expressions and how to apply it to solve problems.

Q&A Guide

Q: What is factoring?

A: Factoring is the process of expressing a quadratic expression as a product of two binomials.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two binomials whose product is equal to the original expression. You can use the following steps:

Find the factors of the constant term.
Find the factors of the coefficient of the linear term.
Write the factored form.

Q: What are the factors of a quadratic expression?

A: The factors of a quadratic expression are the two binomials whose product is equal to the original expression.

Q: How do I find the factors of a quadratic expression?

A: To find the factors of a quadratic expression, 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.

Q: What is the difference between factoring and simplifying?

A: Factoring involves expressing a quadratic expression as a product of two binomials, while simplifying involves combining like terms to reduce the complexity of an expression.

Q: Can I factor a quadratic expression that has a negative coefficient?

A: Yes, you can factor a quadratic expression that has a negative coefficient. The process is the same as factoring a quadratic expression with a positive coefficient.

Q: Can I factor a quadratic expression that has a zero coefficient?

A: Yes, you can factor a quadratic expression that has a zero coefficient. The process is the same as factoring a quadratic expression with a non-zero coefficient.

Q: How do I factor a quadratic expression that has a variable coefficient?

A: To factor a quadratic expression that has a variable coefficient, you need to find two binomials whose product is equal to the original expression. You can use the same steps as factoring a quadratic expression with a constant coefficient.

Q: Can I factor a quadratic expression that has a complex coefficient?

A: Yes, you can factor a quadratic expression that has a complex coefficient. The process is the same as factoring a quadratic expression with a real coefficient.

Q: How do I factor a quadratic expression that has a rational coefficient?

A: To factor a quadratic expression that has a rational coefficient, you need to find two binomials whose product is equal to the original expression. You can use the same steps as factoring a quadratic expression with a real coefficient.

Q: Can I factor a quadratic expression that has a non-rational coefficient?

A: Yes, you can factor a quadratic expression that has a non-rational coefficient. The process is the same as factoring a quadratic expression with a rational coefficient.

Example Problems

Problem 1: Factor the quadratic expression $x^2 + 5x + 6$

To factor the quadratic expression $x^2 + 5x + 6$ , we need to find two numbers whose product is equal to $6$ and whose sum is equal to $5$ .

The factors of $6$ are: $1, 6$ , $2, 3$ , $-1, -6$ , $-2, -3$ , $-6, -1$ , $-3, -2$ , and $-6, -1$ .

The factors of $6$ that add up to $5$ are: $2$ and $3$ .

The factored form of the quadratic expression $x^2 + 5x + 6$ is:

\boxed{(x + 2)(x + 3)}

Problem 2: Factor the quadratic expression $x^2 - 7x - 18$

To factor the quadratic expression $x^2 - 7x - 18$ , we need to find two numbers whose product is equal to $-18$ and whose sum is equal to $-7$ .

The factors of $-18$ are: $1, -18$ , $2, -9$ , $3, -6$ , $-1, 18$ , $-2, 9$ , $-3, 6$ , $-6, 3$ , $-9, 2$ , and $-18, 1$ .

The factors of $-18$ that add up to $-7$ are: $-9$ and $2$ .

The factored form of the quadratic expression $x^2 - 7x - 18$ is:

\boxed{(x - 9)(x + 2)}

Practice Problems

Problem 1: Factor the quadratic expression $x^2 + 3x - 4$

Problem 2: Factor the quadratic expression $x^2 - 2x - 15$

Problem 3: Factor the quadratic expression $x^2 + 5x + 4$

Problem 4: Factor the quadratic expression $x^2 - 7x - 12$

Problem 5: Factor the quadratic expression $x^2 + 2x - 15$

Conclusion

Factoring quadratic expressions is a powerful technique that involves expressing a quadratic equation in the form of a product of two binomials. By following the steps outlined above, we can factor the quadratic expression $x^2 - 2x - 24$ and write it in its factored form. This technique is essential in solving quadratic equations, simplifying expressions, and understanding the properties of quadratic functions. We hope that this Q&A guide has helped you understand the concept of factoring quadratic expressions and how to apply it to solve problems.

Frequently Asked Questions

Q: What is the difference between factoring and simplifying?

A: Factoring involves expressing a quadratic expression as a product of two binomials, while simplifying involves combining like terms to reduce the complexity of an expression.

Q: Can I factor a quadratic expression that has a negative coefficient?

A: Yes, you can factor a quadratic expression that has a negative coefficient. The process is the same as factoring a quadratic expression with a positive coefficient.

Q: Can I factor a quadratic expression that has a zero coefficient?

A: Yes, you can factor a quadratic expression that has a zero coefficient. The process is the same as factoring a quadratic expression with a non-zero coefficient.

Q: How do I factor a quadratic expression that has a variable coefficient?

Q: Can I factor a quadratic expression that has a complex coefficient?

A: Yes, you can factor a quadratic expression that has a complex coefficient. The process is the same as factoring a quadratic expression with a real coefficient.

Q: How do I factor a quadratic expression that has a rational coefficient?

Q: Can I factor a quadratic expression that has a non-rational coefficient?

A: Yes, you can factor a quadratic expression that has a non-rational coefficient. The process is the same as factoring a quadratic expression with a rational coefficient.

Additional Resources

Online Resources

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

Textbooks

"Algebra and Trigonometry" by Michael Sullivan
"College Algebra" by James Stewart
"Precalculus" by Michael Sullivan

Videos

"Factoring Quadratic Expressions" by Khan Academy
"Factoring Quadratic Expressions" by Mathway
"Factoring Quadratic Expressions" by Wolfram Alpha

Introduction

What is Factoring?

The Factorization Process

Step 1: Find the Factors of the Constant Term

Step 2: Find the Factors of the Coefficient of the Linear Term

Step 3: Write the Factored Form

Factoring the Quadratic Expression

Step 1: Find the Factors of the Constant Term

Step 2: Find the Factors of the Coefficient of the Linear Term

Step 3: Write the Factored Form

Conclusion

Frequently Asked Questions

Q: What is factoring?

Q: How do I factor a quadratic expression?

Q: What are the factors of a quadratic expression?

Q: How do I find the factors of a quadratic expression?

Example Problems

Problem 1: Factor the quadratic expression x2+5x+6x^2 + 5x + 6x2+5x+6

Problem 2: Factor the quadratic expression x2−7x−18x^2 - 7x - 18x2−7x−18

Practice Problems

Problem 1: Factor the quadratic expression x2+3x−4x^2 + 3x - 4x2+3x−4

Problem 2: Factor the quadratic expression x2−2x−15x^2 - 2x - 15x2−2x−15

Problem 3: Factor the quadratic expression x2+5x+4x^2 + 5x + 4x2+5x+4

Problem 4: Factor the quadratic expression x2−7x−12x^2 - 7x - 12x2−7x−12

Problem 5: Factor the quadratic expression x2+2x−15x^2 + 2x - 15x2+2x−15

Conclusion

Introduction

Q&A Guide

Q: What is factoring?

Q: How do I factor a quadratic expression?

Q: What are the factors of a quadratic expression?

Q: How do I find the factors of a quadratic expression?

Q: What is the difference between factoring and simplifying?

Q: Can I factor a quadratic expression that has a negative coefficient?

Q: Can I factor a quadratic expression that has a zero coefficient?

Q: How do I factor a quadratic expression that has a variable coefficient?

Q: Can I factor a quadratic expression that has a complex coefficient?

Q: How do I factor a quadratic expression that has a rational coefficient?

Q: Can I factor a quadratic expression that has a non-rational coefficient?

Example Problems

Problem 1: Factor the quadratic expression x2+5x+6x^2 + 5x + 6x2+5x+6

Problem 2: Factor the quadratic expression x2−7x−18x^2 - 7x - 18x2−7x−18

Practice Problems

Problem 1: Factor the quadratic expression x2+3x−4x^2 + 3x - 4x2+3x−4

Problem 2: Factor the quadratic expression x2−2x−15x^2 - 2x - 15x2−2x−15

Problem 3: Factor the quadratic expression x2+5x+4x^2 + 5x + 4x2+5x+4

Problem 4: Factor the quadratic expression x2−7x−12x^2 - 7x - 12x2−7x−12

Problem 5: Factor the quadratic expression x2+2x−15x^2 + 2x - 15x2+2x−15

Conclusion

Frequently Asked Questions

Q: What is the difference between factoring and simplifying?

Q: Can I factor a quadratic expression that has a negative coefficient?

Q: Can I factor a quadratic expression that has a zero coefficient?

Q: How do I factor a quadratic expression that has a variable coefficient?

Q: Can I factor a quadratic expression that has a complex coefficient?

Q: How do I factor a quadratic expression that has a rational coefficient?

Q: Can I factor a quadratic expression that has a non-rational coefficient?

Additional Resources

Online Resources

Textbooks

Videos

Problem 1: Factor the quadratic expression $x^2 + 5x + 6$

Problem 2: Factor the quadratic expression $x^2 - 7x - 18$

Problem 1: Factor the quadratic expression $x^2 + 3x - 4$

Problem 2: Factor the quadratic expression $x^2 - 2x - 15$

Problem 3: Factor the quadratic expression $x^2 + 5x + 4$

Problem 4: Factor the quadratic expression $x^2 - 7x - 12$

Problem 5: Factor the quadratic expression $x^2 + 2x - 15$

Problem 1: Factor the quadratic expression $x^2 + 5x + 6$

Problem 2: Factor the quadratic expression $x^2 - 7x - 18$

Problem 1: Factor the quadratic expression $x^2 + 3x - 4$

Problem 2: Factor the quadratic expression $x^2 - 2x - 15$

Problem 3: Factor the quadratic expression $x^2 + 5x + 4$

Problem 4: Factor the quadratic expression $x^2 - 7x - 12$

Problem 5: Factor the quadratic expression $x^2 + 2x - 15$