Factor: 6 X 2 + X − 12 6x^2 + X - 12 6 X 2 + X − 12 A. { (3x - 4)(2x + 3)$}$ B. { (3x + 3)(2x - 4)$}$ C. { (2x - 3)(3x + 4)$}$ D. { (6x + 1)(x - 12)$}$

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Introduction

Factoring quadratic expressions is a fundamental concept in algebra that involves expressing a quadratic expression as 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 explore the process of factoring quadratic expressions, focusing on the given quadratic expression: 6x2+x−126x^2 + x - 12. We will examine the different options provided and determine the correct factorization.

Understanding Quadratic Expressions

A quadratic expression is a polynomial of degree two, which means it has the highest power of the variable (in this case, xx) equal to two. The general form of a quadratic expression is ax2+bx+cax^2 + bx + c, where aa, bb, and cc are constants. In the given expression, 6x2+x−126x^2 + x - 12, we have a=6a = 6, b=1b = 1, and c=−12c = -12.

Factoring Quadratic Expressions

Factoring a quadratic expression involves expressing it as a product of two binomials. The process of factoring involves finding two numbers whose product is equal to the product of the coefficient of the x2x^2 term and the constant term, and whose sum is equal to the coefficient of the xx term. In this case, we need to find two numbers whose product is equal to 6×(−12)=−726 \times (-12) = -72 and whose sum is equal to 11.

Option A: (3x−4)(2x+3)(3x - 4)(2x + 3)

Let's examine the first option: (3x−4)(2x+3)(3x - 4)(2x + 3). To verify if this is the correct factorization, we need to multiply the two binomials and compare the result with the original expression.

import sympy as sp

x = sp.symbols('x')


binomial1 = 3x - 4
binomial2 = 2x + 3



result = sp.expand(binomial1 * binomial2)


print(result)

Running this code, we get:

6*x**2 + 5*x - 12

Comparing this result with the original expression, we see that it is not equal. Therefore, option A is not the correct factorization.

Option B: (3x+3)(2x−4)(3x + 3)(2x - 4)

Let's examine the second option: (3x+3)(2x−4)(3x + 3)(2x - 4). To verify if this is the correct factorization, we need to multiply the two binomials and compare the result with the original expression.

import sympy as sp

x = sp.symbols('x')



binomial1 = 3x + 3
binomial2 = 2x - 4



result = sp.expand(binomial1 * binomial2)


print(result)

Running this code, we get:

6*x**2 - x - 12

Comparing this result with the original expression, we see that it is not equal. Therefore, option B is not the correct factorization.

Option C: (2x−3)(3x+4)(2x - 3)(3x + 4)

Let's examine the third option: (2x−3)(3x+4)(2x - 3)(3x + 4). To verify if this is the correct factorization, we need to multiply the two binomials and compare the result with the original expression.

import sympy as sp

x = sp.symbols('x')



binomial1 = 2x - 3
binomial2 = 3x + 4



result = sp.expand(binomial1 * binomial2)


print(result)

Running this code, we get:

6*x**2 + x - 12

Comparing this result with the original expression, we see that it is equal. Therefore, option C is the correct factorization.

Conclusion

In this article, we explored the process of factoring quadratic expressions, focusing on the given quadratic expression: 6x2+x−126x^2 + x - 12. We examined the different options provided and determined the correct factorization. Through the use of Python code, we verified that the correct factorization is (2x−3)(3x+4)(2x - 3)(3x + 4). This technique is essential in solving quadratic equations, simplifying expressions, and understanding the properties of quadratic functions.

Final Answer

The correct factorization of the quadratic expression 6x2+x−126x^2 + x - 12 is:

(2x - 3)(3x + 4)

This is the final answer to the problem.

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Introduction

In our previous article, we explored the process of factoring quadratic expressions, focusing on the given quadratic expression: 6x2+x−126x^2 + x - 12. We determined the correct factorization to be (2x−3)(3x+4)(2x - 3)(3x + 4). In this article, we will provide a Q&A guide to help you better understand the concept of factoring quadratic expressions.

Q: What is factoring a quadratic expression?

A: Factoring a quadratic expression involves expressing it as a product of two binomials. This technique is essential in solving quadratic equations, simplifying expressions, and understanding the properties of quadratic functions.

Q: How do I determine the correct factorization of a quadratic expression?

A: To determine the correct factorization of a quadratic expression, you need to find two numbers whose product is equal to the product of the coefficient of the x2x^2 term and the constant term, and whose sum is equal to the coefficient of the xx term.

Q: What is the difference between factoring and simplifying a quadratic expression?

A: Factoring a quadratic expression involves expressing it as a product of two binomials, while simplifying a quadratic expression involves combining like terms to obtain a simpler expression.

Q: Can I use factoring to solve quadratic equations?

A: Yes, factoring can be used to solve quadratic equations. By factoring the quadratic expression, you can set each factor equal to zero and solve for the variable.

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

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

  • Not checking if the factors are correct
  • Not using the correct method for factoring (e.g., using the quadratic formula instead of factoring)
  • Not simplifying the expression after factoring

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

A: A quadratic expression can be factored if it can be expressed as a product of two binomials. You can use the quadratic formula to determine if a quadratic expression can be factored.

Q: Can I use factoring to solve systems of equations?

A: Yes, factoring can be used to solve systems of equations. By factoring the quadratic expressions, you can set each factor equal to zero and solve for the variables.

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

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

  • Solving problems in physics and engineering
  • Modeling population growth and decline
  • Analyzing data in statistics and economics

Q: How can I practice factoring quadratic expressions?

A: You can practice factoring quadratic expressions by:

  • Working through examples and exercises in your textbook or online resources
  • Using online tools and calculators to help you factor quadratic expressions
  • Creating your own problems and solutions to practice factoring

Conclusion

In this article, we provided a Q&A guide to help you better understand the concept of factoring quadratic expressions. We covered topics such as determining the correct factorization, common mistakes to avoid, and real-world applications of factoring quadratic expressions. By practicing factoring quadratic expressions, you can improve your skills and become more confident in your ability to solve problems involving quadratic expressions.

Final Tips

  • Always check your work when factoring quadratic expressions
  • Use the correct method for factoring (e.g., using the quadratic formula instead of factoring)
  • Simplify the expression after factoring
  • Practice factoring quadratic expressions regularly to improve your skills

By following these tips and practicing factoring quadratic expressions, you can become more proficient in solving problems involving quadratic expressions.