Which Statement About $6x^2 + 7x - 10$ Is True?A. One Of The Factors Is $(x + 2)$.B. One Of The Factors Is \$(3x - 2)$[/tex\].C. One Of The Factors Is $(2x + 5)$.D. One Of The Factors Is $(x - 5)$.
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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 the highest power of the variable is two. Factoring quadratic expressions involves breaking them down into simpler expressions, called factors, that can be multiplied together to give the original expression.
Understanding the Problem
The problem asks us to determine which statement about the quadratic expression $6x^2 + 7x - 10$ is true. We are given four options, each stating that one of the factors is a specific expression. To solve this problem, we need to factor the quadratic expression and see which of the given options match one of the factors.
Factoring the Quadratic Expression
To factor the quadratic expression $6x^2 + 7x - 10$, we need to find two numbers whose product is $6 \times (-10) = -60$ and whose sum is $7$. These numbers are $15$ and $-4$, since $15 \times (-4) = -60$ and $15 + (-4) = 11$, but we need a sum of 7, so we need to find the correct pair of numbers.
After reevaluating, we find that the correct pair of numbers is $20$ and $-3$, since $20 \times (-3) = -60$ and $20 + (-3) = 17$, but we need a sum of 7, so we need to find the correct pair of numbers.
After reevaluating, we find that the correct pair of numbers is $15$ and $-4$ is not correct, but we can try to factor the expression by grouping.
Factoring by Grouping
We can factor the quadratic expression $6x^2 + 7x - 10$ by grouping the terms as follows:
Now, we can factor out the greatest common factor (GCF) from each group:
We can now factor out the GCF from each group:
Analyzing the Options
Now that we have factored the quadratic expression, we can analyze the options to see which one matches one of the factors.
Option A states that one of the factors is $(x + 2)$. We can see that this is indeed one of the factors, since we factored the expression as $(x + 2)(6x + 5)$.
Option B states that one of the factors is $(3x - 2)$. We can see that this is not one of the factors, since we factored the expression as $(x + 2)(6x + 5)$.
Option C states that one of the factors is $(2x + 5)$. We can see that this is not one of the factors, since we factored the expression as $(x + 2)(6x + 5)$.
Option D states that one of the factors is $(x - 5)$. We can see that this is not one of the factors, since we factored the expression as $(x + 2)(6x + 5)$.
Conclusion
In conclusion, the correct answer is Option A, which states that one of the factors is $(x + 2)$. This is because we factored the quadratic expression $6x^2 + 7x - 10$ as $(x + 2)(6x + 5)$, and $(x + 2)$ is indeed one of the factors.
Final Answer
The final answer is:
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Introduction
In our previous article, we explored the concept of factoring quadratic expressions and applied it to the quadratic expression $6x^2 + 7x - 10$. We learned how to factor the expression by grouping and identified one of the factors as $(x + 2)$. In this article, we will delve deeper into the world of quadratic expressions and provide a comprehensive guide to factoring and solving them.
Q&A: Factoring Quadratic Expressions
Q: What is a quadratic expression?
A: A quadratic expression is a polynomial of degree two, which means the highest power of the variable is two. It can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.
Q: How do I factor a quadratic expression?
A: There are several methods to factor a quadratic expression, including:
- Factoring by grouping
- Factoring by using the greatest common factor (GCF)
- Factoring by using the difference of squares formula
- Factoring by using the perfect square trinomial formula
Q: What is factoring by grouping?
A: Factoring by grouping involves grouping the terms of the quadratic expression into two groups and then factoring out the greatest common factor (GCF) from each group.
Q: What is the greatest common factor (GCF)?
A: The greatest common factor (GCF) is the largest factor that divides each term of the quadratic expression.
Q: How do I factor a quadratic expression using the difference of squares formula?
A: The difference of squares formula is $a^2 - b^2 = (a + b)(a - b)$. To factor a quadratic expression using this formula, you need to identify two perfect squares that differ by the constant term.
Q: How do I factor a quadratic expression using the perfect square trinomial formula?
A: The perfect square trinomial formula is $(a + b)^2 = a^2 + 2ab + b^2$ or $(a - b)^2 = a^2 - 2ab + b^2$. To factor a quadratic expression using this formula, you need to identify two perfect squares that differ by the constant term.
Q: What is the difference between factoring and solving a quadratic expression?
A: Factoring a quadratic expression involves breaking it down into simpler expressions, called factors, that can be multiplied together to give the original expression. Solving a quadratic expression involves finding the values of the variable that make the expression equal to zero.
Q: How do I solve a quadratic expression?
A: To solve a quadratic expression, you need to set the expression equal to zero and then use algebraic methods to find the values of the variable that satisfy the equation.
Conclusion
In conclusion, factoring and solving quadratic expressions are essential skills in algebra that can be applied to a wide range of problems. By understanding the different methods of factoring and solving quadratic expressions, you can tackle complex problems with confidence.
Final Tips
- Practice, practice, practice: The more you practice factoring and solving quadratic expressions, the more comfortable you will become with the different methods.
- Use algebraic methods: Algebraic methods, such as factoring and solving, are essential tools for solving quadratic expressions.
- Be patient: Factoring and solving quadratic expressions can be challenging, so be patient and take your time.
Final Answer
The final answer is: There is no final answer, as this article is a comprehensive guide to factoring and solving quadratic expressions.