Factor Completely:$\[ 2x^2 + 8x + 6 \\]A. $\[ 2(x+3)(x+1) \\]B. $\[ (x+3)(x+1) \\]C. $\[ 2(x+3)(x-1) \\]D. $\[ (x-3)(x-1) \\]
Understanding the Problem
Factoring a quadratic expression involves expressing it as a product of two binomials. In this problem, we are given the quadratic expression 2x^2 + 8x + 6, and we need to factor it completely. This means we need to express it as a product of two binomials in the form (ax + b)(cx + d).
Step 1: Look for Common Factors
The first step in factoring a quadratic expression is to look for any common factors. In this case, we can see that the expression 2x^2 + 8x + 6 has a common factor of 2. We can factor out 2 from each term:
2x^2 + 8x + 6 = 2(x^2 + 4x + 3)
Step 2: Look for Two Binomials
Now that we have factored out 2, we need to look for two binomials that multiply together to give us the quadratic expression x^2 + 4x + 3. We can start by looking for two numbers whose product is 3 and whose sum is 4. These numbers are 1 and 3, so we can write:
x^2 + 4x + 3 = (x + 1)(x + 3)
Step 3: Write the Final Factored Form
Now that we have found the two binomials, we can write the final factored form of the expression:
2x^2 + 8x + 6 = 2(x + 1)(x + 3)
Conclusion
In this problem, we factored the quadratic expression 2x^2 + 8x + 6 completely by first factoring out 2, and then looking for two binomials that multiply together to give us the quadratic expression x^2 + 4x + 3. We found that the two binomials are (x + 1) and (x + 3), and we wrote the final factored form of the expression as 2(x + 1)(x + 3).
Comparison with Answer Choices
Now that we have factored the expression completely, we can compare our answer with the answer choices. We can see that our answer, 2(x + 1)(x + 3), matches answer choice A.
Answer Choice Comparison
| Answer Choice | Factored Form |
|---|---|
| A | 2(x + 3)(x + 1) |
| B | (x + 3)(x + 1) |
| C | 2(x + 3)(x - 1) |
| D | (x - 3)(x - 1) |
As we can see, our answer matches answer choice A, which is 2(x + 3)(x + 1).
Final Answer
Understanding the Problem
Factoring a quadratic expression involves expressing it as a product of two binomials. In this problem, we are given the quadratic expression 2x^2 + 8x + 6, and we need to factor it completely. This means we need to express it as a product of two binomials in the form (ax + b)(cx + d).
Q&A Session
Q: What is the first step in factoring a quadratic expression?
A: The first step in factoring a quadratic expression is to look for any common factors. In this case, we can see that the expression 2x^2 + 8x + 6 has a common factor of 2. We can factor out 2 from each term:
2x^2 + 8x + 6 = 2(x^2 + 4x + 3)
Q: How do I find the two binomials that multiply together to give me the quadratic expression?
A: To find the two binomials, we need to look for two numbers whose product is 3 and whose sum is 4. These numbers are 1 and 3, so we can write:
x^2 + 4x + 3 = (x + 1)(x + 3)
Q: What is the final factored form of the expression?
A: Now that we have found the two binomials, we can write the final factored form of the expression:
2x^2 + 8x + 6 = 2(x + 1)(x + 3)
Q: How do I compare my answer with the answer choices?
A: To compare your answer with the answer choices, you can look at the factored form of each answer choice and see which one matches your answer. In this case, our answer, 2(x + 1)(x + 3), matches answer choice A.
Q: What is the final answer?
A: The final answer is .
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 factoring out any common factors
- Not looking for two binomials that multiply together to give the quadratic expression
- Not writing the final factored form of the expression correctly
Q: How do I practice factoring quadratic expressions?
A: To practice factoring quadratic expressions, you can try factoring different quadratic expressions on your own. You can also use online resources or practice problems to help you practice factoring quadratic expressions.
Q: What are some real-world applications of factoring quadratic expressions?
A: Factoring quadratic expressions has many real-world applications, including:
- Solving systems of equations
- Finding the maximum or minimum value of a function
- Modeling real-world situations with quadratic equations
Q: How do I know if I have factored a quadratic expression correctly?
A: To know if you have factored a quadratic expression correctly, you can check your answer by multiplying the two binomials together and seeing if you get the original quadratic expression. You can also use online resources or check your answer with a calculator.
Q: What are some tips for factoring quadratic expressions?
A: Some tips for factoring quadratic expressions include:
- Looking for common factors first
- Using the distributive property to expand the expression
- Checking your answer by multiplying the two binomials together
Q: How do I factor a quadratic expression with a negative coefficient?
A: To factor a quadratic expression with a negative coefficient, you can use the same steps as factoring a quadratic expression with a positive coefficient. However, you will need to use a negative sign in one of the binomials.
Q: What are some common quadratic expressions that can be factored?
A: Some common quadratic expressions that can be factored include:
- x^2 + 4x + 4
- x^2 - 4x + 4
- x^2 + 2x - 6
- x^2 - 2x - 15
Q: How do I factor a quadratic expression with a coefficient of 1?
A: To factor a quadratic expression with a coefficient of 1, you can use the same steps as factoring a quadratic expression with a coefficient other than 1. However, you will not need to factor out any common factors.
Q: What are some real-world examples of factoring quadratic expressions?
A: Some real-world examples of factoring quadratic expressions include:
- Modeling the height of a projectile as a function of time
- Finding the maximum or minimum value of a function
- Solving systems of equations
Q: How do I factor a quadratic expression with a coefficient of 0?
A: To factor a quadratic expression with a coefficient of 0, you can use the same steps as factoring a quadratic expression with a coefficient other than 0. However, you will need to factor out any common factors.
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 any common factors
- Not looking for two binomials that multiply together to give the quadratic expression
- Not writing the final factored form of the expression correctly
Q: How do I practice factoring quadratic expressions?
A: To practice factoring quadratic expressions, you can try factoring different quadratic expressions on your own. You can also use online resources or practice problems to help you practice factoring quadratic expressions.
Q: What are some real-world applications of factoring quadratic expressions?
A: Factoring quadratic expressions has many real-world applications, including:
- Solving systems of equations
- Finding the maximum or minimum value of a function
- Modeling real-world situations with quadratic equations
Q: How do I know if I have factored a quadratic expression correctly?
A: To know if you have factored a quadratic expression correctly, you can check your answer by multiplying the two binomials together and seeing if you get the original quadratic expression. You can also use online resources or check your answer with a calculator.
Q: What are some tips for factoring quadratic expressions?
A: Some tips for factoring quadratic expressions include:
- Looking for common factors first
- Using the distributive property to expand the expression
- Checking your answer by multiplying the two binomials together
Q: How do I factor a quadratic expression with a negative coefficient?
A: To factor a quadratic expression with a negative coefficient, you can use the same steps as factoring a quadratic expression with a positive coefficient. However, you will need to use a negative sign in one of the binomials.
Q: What are some common quadratic expressions that can be factored?
A: Some common quadratic expressions that can be factored include:
- x^2 + 4x + 4
- x^2 - 4x + 4
- x^2 + 2x - 6
- x^2 - 2x - 15
Q: How do I factor a quadratic expression with a coefficient of 1?
A: To factor a quadratic expression with a coefficient of 1, you can use the same steps as factoring a quadratic expression with a coefficient other than 1. However, you will not need to factor out any common factors.
Q: What are some real-world examples of factoring quadratic expressions?
A: Some real-world examples of factoring quadratic expressions include:
- Modeling the height of a projectile as a function of time
- Finding the maximum or minimum value of a function
- Solving systems of equations
Q: How do I factor a quadratic expression with a coefficient of 0?
A: To factor a quadratic expression with a coefficient of 0, you can use the same steps as factoring a quadratic expression with a coefficient other than 0. However, you will need to factor out any common factors.
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 any common factors
- Not looking for two binomials that multiply together to give the quadratic expression
- Not writing the final factored form of the expression correctly
Q: How do I practice factoring quadratic expressions?
A: To practice factoring quadratic expressions, you can try factoring different quadratic expressions on your own. You can also use online resources or practice problems to help you practice factoring quadratic expressions.
Q: What are some real-world applications of factoring quadratic expressions?
A: Factoring quadratic expressions has many real-world applications, including:
- Solving systems of equations
- Finding the maximum or minimum value of a function
- Modeling real-world situations with quadratic equations
Q: How do I know if I have factored a quadratic expression correctly?
A: To know if you have factored a quadratic expression correctly, you can check your answer by multiplying the two binomials together and seeing if you get the original quadratic expression. You can also use online resources or check your answer with a calculator.
Q: What are some tips for factoring quadratic expressions?
A: Some tips for factoring quadratic expressions include:
- Looking for common factors first
- Using the distributive property to expand the expression
- Checking your answer by multiplying the two binomials together
Q: How do I factor a quadratic expression with a negative coefficient?
A: To factor a quadratic expression with a negative coefficient, you can use the same steps as factoring a quadratic expression with