Which Is The Factorization Of $8x^2 + 13x - 6$?A. $(8x - 3)(x + 2)$ B. \$(x - 6)(8x + 1)$[/tex\] C. $(2x - 2)(4x + 3)$ D. $(4x - 1)(2x + 6)$

Feb 27, 2025 by ADMIN 155 views

**Factorization of Quadratic Expressions: A Step-by-Step Guide**

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Introduction

In algebra, factorization is a crucial concept that helps us simplify complex expressions and solve equations. A quadratic expression is a polynomial of degree two, which means it has a highest power of two. Factorizing a quadratic expression involves expressing it as a product of two binomials. In this article, we will explore the factorization of the quadratic expression $8x^2 + 13x - 6$ and determine the correct factorization among the given options.

Understanding the Factorization Process

To factorize a quadratic expression, we need to find two binomials whose product equals the original expression. The general form of a quadratic expression is $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. We can factorize a quadratic expression by finding two numbers whose product is $ac$ and whose sum is $b$ . These numbers are the roots of the quadratic equation $ax^2 + bx + c = 0$ .

The Factorization of $8x^2 + 13x - 6$

To factorize the quadratic expression $8x^2 + 13x - 6$ , we need to find two binomials whose product equals the original expression. Let's start by finding the factors of the constant term $-6$ . The factors of $-6$ are $\pm 1, \pm 2, \pm 3, \pm 6$ . We need to find two numbers whose product is $-6$ and whose sum is $13$ . After some trial and error, we find that the numbers $8$ and $-1$ satisfy these conditions.

Factoring by Grouping

We can factorize the quadratic expression $8x^2 + 13x - 6$ by grouping the terms. We can write the expression as $(8x^2 + 6x) + (7x - 6)$ . Now, we can factor out the greatest common factor (GCF) from each group. The GCF of $8x^2 + 6x$ is $2x$ , and the GCF of $7x - 6$ is $1$ . Therefore, we can write the expression as $2x(4x + 3) + 1(7x - 6)$ .

Factoring by Using the AC Method

Another method to factorize a quadratic expression is by using the AC method. This method involves finding the factors of the product $ac$ and then finding the binomials whose product equals the original expression. In this case, the product $ac$ is $8 \times -6 = -48$ . We need to find two numbers whose product is $-48$ and whose sum is $13$ . After some trial and error, we find that the numbers $8$ and $-6$ satisfy these conditions.

Evaluating the Options

Now that we have factorized the quadratic expression $8x^2 + 13x - 6$ , we can evaluate the given options. The correct factorization is $(8x - 3)(x + 2)$ .

Conclusion

In conclusion, factorization is a crucial concept in algebra that helps us simplify complex expressions and solve equations. We have explored the factorization of the quadratic expression $8x^2 + 13x - 6$ and determined the correct factorization among the given options. By using the factorization process, we can simplify complex expressions and solve equations more efficiently.

Final Answer

The final answer is: $\boxed{A}$

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Linear Algebra" by Jim Hefferon

Additional Resources

Khan Academy: Factorization of Quadratic Expressions
MIT OpenCourseWare: Algebra
Wolfram Alpha: Factorization of Quadratic Expressions

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Introduction

In our previous article, we explored the factorization of the quadratic expression $8x^2 + 13x - 6$ and determined the correct factorization among the given options. In this article, we will answer some common questions related to quadratic factorization.

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 rewriting it in a more compact form.

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 binomials whose product equals the original expression. You can use the factorization process, which involves finding the factors of the constant term and the sum of the coefficients.

Q: What is the AC method in quadratic factorization?

A: The AC method is a technique used to factorize a quadratic expression by finding the factors of the product $ac$ and then finding the binomials whose product equals the original expression.

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

A: Yes, you can factorize a quadratic expression with a negative leading coefficient. In this case, you need to find two binomials whose product equals the original expression and whose signs are opposite.

Q: How do I factorize a quadratic expression with a coefficient of 1?

A: To factorize a quadratic expression with a coefficient of 1, you need to find two binomials whose product equals the original expression and whose coefficients are 1.

Q: Can I use the factorization process to factorize a quadratic expression with a coefficient of 0?

A: No, you cannot use the factorization process to factorize a quadratic expression with a coefficient of 0. In this case, the expression is already in its simplest form.

Q: What is the difference between factoring and solving a quadratic equation?

A: Factoring a quadratic equation involves expressing it as a product of two binomials, while solving a quadratic equation involves finding the values of the variable that satisfy the equation.

Q: Can I use the factorization process to solve a quadratic equation?

A: Yes, you can use the factorization process to solve a quadratic equation. By factoring the equation, you can find the values of the variable that satisfy the equation.

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

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

Not finding the correct factors of the constant term
Not finding the correct binomials whose product equals the original expression
Not checking the signs of the binomials
Not simplifying the expression after factoring

Conclusion

In conclusion, quadratic factorization is a crucial concept in algebra that helps us simplify complex expressions and solve equations. We have answered some common questions related to quadratic factorization and provided some tips and tricks to help you master this concept.

Final Answer

The final answer is: $\boxed{A}$

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Linear Algebra" by Jim Hefferon

Additional Resources

Khan Academy: Factorization of Quadratic Expressions
MIT OpenCourseWare: Algebra
Wolfram Alpha: Factorization of Quadratic Expressions

Introduction

Understanding the Factorization Process

The Factorization of 8x2+13x−68x^2 + 13x - 68x2+13x−6

Factoring by Grouping

Factoring by Using the AC Method

Evaluating the Options

Conclusion

Final Answer

References

Additional Resources

Introduction

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

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

Q: What is the AC method in quadratic factorization?

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

Q: How do I factorize a quadratic expression with a coefficient of 1?

Q: Can I use the factorization process to factorize a quadratic expression with a coefficient of 0?

Q: What is the difference between factoring and solving a quadratic equation?

Q: Can I use the factorization process to solve a quadratic equation?

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

Conclusion

Final Answer

References

Additional Resources

The Factorization of $8x^2 + 13x - 6$