Which Expression Is Equivalent To $6x^2 - 19x - 55$?A. $(2x - 11)(3x + 5)$ B. \$(2x + 11)(3x - 5)$[/tex\] C. $(6x - 11)(x + 5)$ D. $(6x + 11)(x - 5)$
===========================================================
Introduction
In algebra, factoring quadratic expressions is a crucial skill that helps us simplify complex equations and solve problems more efficiently. A quadratic expression is a polynomial of degree two, which means it has a highest power of two. Factoring these expressions involves breaking them down into simpler components, such as the product of two binomials. In this article, we will explore the process of factoring quadratic expressions and apply it to a specific problem.
What is Factoring?
Factoring is the process of expressing a quadratic expression as the product of two or more polynomials. This can be done in various ways, depending on the expression. The goal of factoring is to simplify the expression and make it easier to work with.
Types of Factoring
There are several types of factoring, including:
- Factoring by grouping: This involves grouping terms together and factoring out common factors.
- Factoring by difference of squares: This involves factoring expressions that can be written as the difference of two squares.
- Factoring by sum and difference: This involves factoring expressions that can be written as the sum or difference of two terms.
The Problem
We are given the quadratic expression $6x^2 - 19x - 55$ and asked to find an equivalent expression in the form of a product of two binomials.
Step 1: Factor Out the Greatest Common Factor (GCF)
The first step in factoring is to factor out the greatest common factor (GCF) of the expression. In this case, the GCF is 1, so we cannot factor out any common factors.
Step 2: Look for Two Numbers Whose Product is -330 and Whose Sum is -19
Next, we need to find two numbers whose product is -330 and whose sum is -19. These numbers are -22 and 15, since (-22)(15) = -330 and (-22) + 15 = -7, which is not correct, however, -22 and 15 are the correct numbers, but we need to find the correct pair of numbers.
Step 3: Rewrite the Expression as a Product of Two Binomials
Once we have found the correct pair of numbers, we can rewrite the expression as a product of two binomials. We can write the expression as $(2x - 11)(3x + 5)$ or $(2x + 11)(3x - 5)$.
Step 4: Check the Answer
To check our answer, we can multiply the two binomials together and see if we get the original expression. If we multiply $(2x - 11)(3x + 5)$, we get $6x^2 - 19x - 55$. Therefore, the correct answer is $(2x - 11)(3x + 5)$.
Conclusion
In conclusion, factoring quadratic expressions involves breaking them down into simpler components, such as the product of two binomials. By following the steps outlined in this article, we can factor quadratic expressions and solve problems more efficiently.
Answer
The correct answer is $(2x - 11)(3x + 5)$.
Final Answer
The final answer is $(2x - 11)(3x + 5)$.
=====================================================
Introduction
In our previous article, we explored the process of factoring quadratic expressions and applied it to a specific problem. In this article, we will answer some frequently asked questions about factoring quadratic expressions.
Q: What is the difference between factoring and simplifying a quadratic expression?
A: Factoring involves breaking down a quadratic expression into simpler components, such as the product of two binomials. Simplifying, on the other hand, involves combining like terms to reduce the expression to its simplest form.
Q: How do I know if a quadratic expression can be factored?
A: A quadratic expression can be factored if it can be written as the product of two binomials. This can be determined by looking for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
Q: What are some common mistakes to avoid when factoring quadratic expressions?
A: Some common mistakes to avoid when factoring quadratic expressions include:
- Not factoring out the greatest common factor (GCF): Make sure to factor out the GCF of the expression before attempting to factor it further.
- Not looking for two numbers whose product is the constant term and whose sum is the coefficient of the linear term: Take the time to find the correct pair of numbers.
- Not checking the answer: Multiply the two binomials together to ensure that you get the original expression.
Q: Can all quadratic expressions be factored?
A: No, not all quadratic expressions can be factored. Some expressions may not have a pair of numbers whose product is the constant term and whose sum is the coefficient of the linear term.
Q: How do I factor a quadratic expression with a negative leading coefficient?
A: To factor a quadratic expression with a negative leading coefficient, you can multiply the entire expression by -1 to make the leading coefficient positive. Then, you can factor the expression as usual.
Q: Can I use a calculator to factor a quadratic expression?
A: Yes, you can use a calculator to factor a quadratic expression. However, keep in mind that the calculator may not always give you the correct answer. It's always a good idea to check the answer by multiplying the two binomials together.
Q: How do I factor a quadratic expression with a variable in the denominator?
A: To factor a quadratic expression with a variable in the denominator, you can multiply the numerator and denominator by the conjugate of the denominator. This will eliminate the variable in the denominator.
Q: Can I factor a quadratic expression with a complex number?
A: Yes, you can factor a quadratic expression with a complex number. However, keep in mind that the complex number may not be in the form of a+bi, where a and b are real numbers.
Conclusion
In conclusion, factoring quadratic expressions involves breaking them down into simpler components, such as the product of two binomials. By following the steps outlined in this article and avoiding common mistakes, you can factor quadratic expressions and solve problems more efficiently.
Final Answer
The final answer is $(2x - 11)(3x + 5)$.