Factoring Quadratic Expression Find The Value Of B

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#h1 Factoring Quadratic Expressions: Finding the Correct Value of b

In mathematics, factoring quadratic expressions is a fundamental skill, especially when solving quadratic equations and simplifying algebraic expressions. This article delves into the process of factoring a specific quadratic expression, x^2 + bx + 18, and aims to identify the value of 'b' that allows for successful factorization. Understanding the relationship between the coefficients of a quadratic expression and its factors is crucial for mastering this concept. Let's embark on this mathematical journey to unravel the mystery of 'b' and its role in factoring quadratic expressions.

Understanding Quadratic Expressions and Factoring

To effectively determine the correct value of 'b', it's essential to first grasp the basics of quadratic expressions and the process of factoring. A quadratic expression is a polynomial of degree two, generally written in the form ax^2 + bx + c, where 'a', 'b', and 'c' are constants, and 'x' is a variable. Factoring a quadratic expression involves breaking it down into a product of two binomials. In simpler terms, we seek to rewrite the expression in the form (x + p)(x + q), where 'p' and 'q' are constants.

When we expand (x + p)(x + q), we get x^2 + (p + q)x + pq. By comparing this with the general form ax^2 + bx + c, we can observe a direct relationship between the coefficients: 'b' corresponds to the sum of 'p' and 'q' (i.e., b = p + q), and 'c' corresponds to the product of 'p' and 'q' (i.e., c = pq). This relationship is the cornerstone of factoring quadratic expressions. To successfully factor a quadratic expression, we need to find two numbers, 'p' and 'q', that satisfy these conditions. For the given expression, x^2 + bx + 18, we know that c = 18. Therefore, we need to find two numbers whose product is 18 and whose sum is equal to the unknown coefficient 'b'.

Identifying Factor Pairs of 18

The first step in finding the correct value of 'b' is to identify all the factor pairs of 18. Factor pairs are sets of two numbers that, when multiplied together, give the desired product. In this case, we are looking for pairs that multiply to 18. Let's list the factor pairs of 18:

  • 1 and 18
  • 2 and 9
  • 3 and 6
  • -1 and -18
  • -2 and -9
  • -3 and -6

Each of these pairs, when multiplied, results in 18. Now, we need to consider the sums of these pairs to find the possible values of 'b'. Remember that 'b' is the sum of the two numbers in the factor pair. By systematically examining these pairs, we can determine which sum corresponds to one of the answer choices provided.

Calculating Possible Values of 'b'

Now that we have identified the factor pairs of 18, the next step is to calculate the sum of each pair. This will give us the possible values for the coefficient 'b' in the quadratic expression x^2 + bx + 18. Let's go through each pair:

  • 1 + 18 = 19
  • 2 + 9 = 11
  • 3 + 6 = 9
  • -1 + (-18) = -19
  • -2 + (-9) = -11
  • -3 + (-6) = -9

These sums represent all the possible values of 'b' for which the given quadratic expression can be factored using integer coefficients. By examining these values, we can determine which one matches the options provided in the question.

Matching the Calculated Values with the Answer Choices

Now, let's compare the possible values of 'b' we calculated with the answer choices provided:

  • A. -19
  • B. 17
  • C. 7
  • D. 3

From our calculations, we found that -19 is a possible value of 'b' (corresponding to the factor pair -1 and -18). Let's check if any other calculated values match the options. We have 19, 11, 9, -11, and -9 as other possible values for 'b'. However, only -19 is present in the answer choices. Therefore, the correct value of 'b' for which the expression x^2 + bx + 18 can be factored is -19.

Constructing the Factored Form

To further solidify our understanding, let's construct the factored form of the quadratic expression when b = -19. We know that the factor pair corresponding to b = -19 is -1 and -18. Therefore, the factored form of the expression is:

  • (x - 1)(x - 18)

Expanding this expression, we get:

  • x^2 - 18x - x + 18 = x^2 - 19x + 18

This confirms that when b = -19, the expression x^2 + bx + 18 can indeed be factored into (x - 1)(x - 18). This exercise demonstrates the relationship between the value of 'b', the factor pairs of the constant term, and the factored form of the quadratic expression. Understanding this relationship is crucial for mastering factoring techniques and solving quadratic equations.

Conclusion: The Significance of 'b' in Factoring

In conclusion, the value of 'b' plays a pivotal role in determining whether a quadratic expression can be factored. By understanding the relationship between the coefficients of the quadratic expression and its factors, we can effectively identify the correct value of 'b' that allows for successful factorization. In the case of x^2 + bx + 18, the correct value of 'b' is -19, as it corresponds to the factor pair -1 and -18, allowing the expression to be factored into (x - 1)(x - 18). This exercise highlights the importance of mastering factoring techniques in algebra and provides a solid foundation for solving more complex mathematical problems.

#h2 Select the Correct Answer for Factoring Quadratic Expressions

This section focuses on providing a detailed explanation of how to select the correct answer for the question: For which value of b can the expression x^2 + bx + 18 be factored? The answer choices are A. -19, B. 17, C. 7, and D. 3. We will walk through the step-by-step process of finding the right value of 'b' by understanding the principles of factoring quadratic expressions and identifying the correct factor pairs.

Step 1: Understanding the Factoring Principle

The core principle behind factoring a quadratic expression of the form x^2 + bx + c involves finding two numbers, let's call them 'p' and 'q', such that their product equals 'c' and their sum equals 'b'. In other words, we need to find 'p' and 'q' that satisfy the following conditions:

  • p * q = c
  • p + q = b

Once we find these numbers, we can rewrite the quadratic expression in its factored form as (x + p)(x + q). This fundamental concept is the key to solving our problem. In our specific case, the quadratic expression is x^2 + bx + 18, so we know that c = 18. Therefore, we need to find two numbers whose product is 18 and whose sum matches one of the given answer choices for 'b'.

Step 2: Identifying Factor Pairs of 18

The next step is to systematically identify all the pairs of integers that multiply to 18. These pairs are known as factor pairs. Listing all factor pairs ensures that we don't miss any potential values for 'p' and 'q'. The factor pairs of 18 are:

  • 1 and 18
  • 2 and 9
  • 3 and 6
  • -1 and -18
  • -2 and -9
  • -3 and -6

We include both positive and negative pairs because the product of two negative numbers is also positive. This comprehensive list is essential for finding the correct value of 'b'.

Step 3: Calculating the Sum of Each Factor Pair

Now that we have the factor pairs, we need to calculate the sum of each pair. The sum of each pair represents a possible value for 'b'. Let's calculate the sums:

  • 1 + 18 = 19
  • 2 + 9 = 11
  • 3 + 6 = 9
  • -1 + (-18) = -19
  • -2 + (-9) = -11
  • -3 + (-6) = -9

These sums give us the possible values of 'b' that would allow the quadratic expression x^2 + bx + 18 to be factored using integer coefficients.

Step 4: Comparing the Calculated Values with the Answer Choices

The final step is to compare the calculated sums with the answer choices provided in the question. The answer choices are:

  • A. -19
  • B. 17
  • C. 7
  • D. 3

By comparing the calculated sums (19, 11, 9, -19, -11, -9) with the answer choices, we can see that -19 is the only value that matches one of the choices. Therefore, the correct answer is A. -19. This indicates that when b = -19, the quadratic expression x^2 + bx + 18 can be factored.

Step 5: Verifying the Answer by Factoring

To verify our answer, let's factor the quadratic expression with b = -19. The expression becomes x^2 - 19x + 18. We already know that the factor pair corresponding to b = -19 is -1 and -18. So, we can write the expression in its factored form as:

  • (x - 1)(x - 18)

Expanding this factored form, we get:

  • x^2 - 18x - x + 18 = x^2 - 19x + 18

This confirms that the factored form is correct, and our answer is indeed A. -19. The ability to factor quadratic expressions is a critical skill in algebra, and this step-by-step approach ensures accuracy and understanding.

Conclusion: Selecting the Correct Value of 'b'

In conclusion, by understanding the principles of factoring quadratic expressions, identifying factor pairs, calculating their sums, and comparing these sums with the given answer choices, we can confidently select the correct value of 'b'. For the expression x^2 + bx + 18, the correct value of 'b' is -19, which corresponds to the factor pair -1 and -18. This comprehensive method not only provides the correct answer but also reinforces the fundamental concepts of factoring quadratic expressions.

#h2 Detailed Explanation of the Correct Answer: A. -19

This section will provide an in-depth explanation of why option A, -19, is the correct answer for the question: For which value of b can the expression x^2 + bx + 18 be factored? We will revisit the steps involved in factoring quadratic expressions, focusing on the specific factor pair that makes -19 the appropriate value for 'b'. This thorough explanation aims to solidify your understanding of the factoring process and the significance of each step.

Reviewing the Factoring Process

As discussed earlier, factoring a quadratic expression of the form x^2 + bx + c involves finding two numbers, 'p' and 'q', such that their product equals 'c' and their sum equals 'b'. This means:

  • p * q = c
  • p + q = b

For the given expression, x^2 + bx + 18, we know that c = 18. Thus, we need to identify two numbers that multiply to 18 and add up to the value of 'b' that is presented in the answer choices. We have already listed all the factor pairs of 18, which are:

  • 1 and 18
  • 2 and 9
  • 3 and 6
  • -1 and -18
  • -2 and -9
  • -3 and -6

Each of these pairs, when multiplied, results in 18. Our goal is to find the pair whose sum matches one of the answer choices, specifically A. -19.

Identifying the Correct Factor Pair for b = -19

To determine if -19 is a valid value for 'b', we need to find a factor pair of 18 whose sum is -19. Let's examine the factor pairs and their sums again:

  • 1 + 18 = 19
  • 2 + 9 = 11
  • 3 + 6 = 9
  • -1 + (-18) = -19
  • -2 + (-9) = -11
  • -3 + (-6) = -9

From the list, we can clearly see that the factor pair -1 and -18 has a sum of -19. This means that if we choose p = -1 and q = -18, the conditions p * q = 18 and p + q = -19 are satisfied. This is the key to understanding why -19 is the correct answer.

Constructing the Factored Form with b = -19

Now that we have identified the correct factor pair for b = -19, we can write the quadratic expression in its factored form. When b = -19, the expression is x^2 - 19x + 18. Using the factor pair -1 and -18, we can write the factored form as:

  • (x - 1)(x - 18)

This factored form represents the product of two binomials, where the constants -1 and -18 are the numbers we found whose product is 18 and whose sum is -19. To verify that this is correct, we can expand the factored form.

Verifying the Factored Form

Let's expand the factored form (x - 1)(x - 18):

  • (x - 1)(x - 18) = x(x - 18) - 1(x - 18)
  • = x^2 - 18x - x + 18
  • = x^2 - 19x + 18

This expansion results in the original quadratic expression with b = -19, which confirms that our factored form is correct. This step is crucial in ensuring that we have not made any errors in our calculations and that our answer is indeed accurate.

Why the Other Options Are Incorrect

To further solidify our understanding, let's briefly explain why the other options are incorrect. Options B, C, and D are 17, 7, and 3, respectively. We need to check if any factor pairs of 18 add up to these values.

  • For b = 17, there is no factor pair of 18 that adds up to 17.
  • For b = 7, there is no factor pair of 18 that adds up to 7.
  • For b = 3, there is no factor pair of 18 that adds up to 3.

Since none of these values can be obtained by adding any factor pair of 18, they cannot be the correct values of 'b' for which the expression x^2 + bx + 18 can be factored. This comparison reinforces why -19 is the only valid answer.

Conclusion: The Significance of Factor Pairs

In conclusion, the correct answer for the question For which value of b can the expression x^2 + bx + 18 be factored? is A. -19. This is because the factor pair -1 and -18 of 18 adds up to -19, allowing us to factor the expression as (x - 1)(x - 18). This detailed explanation highlights the importance of understanding factor pairs and their sums in the process of factoring quadratic expressions. By systematically identifying factor pairs and verifying the results, we can confidently solve such problems and enhance our algebraic skills.

#h3 Practice Questions for Factoring Quadratic Expressions

To further enhance your understanding of factoring quadratic expressions and the role of the coefficient 'b', let's explore some practice questions. These questions will help you apply the concepts we've discussed and improve your problem-solving skills. Each question involves finding the value of 'b' for which a given quadratic expression can be factored. By working through these examples, you'll become more proficient in identifying factor pairs and applying the principles of factoring.

Practice Question 1

For which value of 'b' can the expression x^2 + bx + 24 be factored?

A. 10

B. 11

C. 14

D. 25

Solution

To solve this question, we need to find two numbers that multiply to 24 and add up to one of the values given in the answer choices. Let's list the factor pairs of 24:

  • 1 and 24

  • 2 and 12

  • 3 and 8

  • 4 and 6

  • -1 and -24

  • -2 and -12

  • -3 and -8

  • -4 and -6

Now, let's find the sums of these pairs:

  • 1 + 24 = 25

  • 2 + 12 = 14

  • 3 + 8 = 11

  • 4 + 6 = 10

  • -1 + (-24) = -25

  • -2 + (-12) = -14

  • -3 + (-8) = -11

  • -4 + (-6) = -10

By comparing these sums with the answer choices, we find that the possible values of 'b' are 25, 14, 11, and 10. These values correspond to choices D, C, B and A respectively. Therefore, all the answer options are valid possible values of b. If a single correct option was sought from the provided choices, the answer would be D. 25, as this value matches one of the calculated sums. The factored form for b = 25 would be (x + 1)(x + 24).

Practice Question 2

What value of 'b' makes the expression x^2 + bx - 15 factorable?

A. 14

B. 8

C. 2

D. 4

Solution

In this case, we need to find two numbers that multiply to -15 and add up to one of the values given in the answer choices. The factor pairs of -15 are:

  • 1 and -15

  • -1 and 15

  • 3 and -5

  • -3 and 5

Now, let's calculate the sums of these pairs:

  • 1 + (-15) = -14

  • -1 + 15 = 14

  • 3 + (-5) = -2

  • -3 + 5 = 2

Comparing these sums with the answer choices, we find that 2 and 14 are possible values of 'b'. If a single correct option was sought from the provided choices, the answer would be C. 2, as this value matches one of the calculated sums. The factored form for b = 2 would be (x - 3)(x + 5).

Practice Question 3

For what value of b can x^2 + bx + 36 be factored?

A. 5

B. 10

C. 12

D. 15

Solution

We need to find two numbers that multiply to 36 and add up to one of the values in the answer choices. The factor pairs of 36 are:

  • 1 and 36

  • 2 and 18

  • 3 and 12

  • 4 and 9

  • 6 and 6

  • -1 and -36

  • -2 and -18

  • -3 and -12

  • -4 and -9

  • -6 and -6

Let's find the sums of these pairs:

  • 1 + 36 = 37

  • 2 + 18 = 20

  • 3 + 12 = 15

  • 4 + 9 = 13

  • 6 + 6 = 12

  • -1 + (-36) = -37

  • -2 + (-18) = -20

  • -3 + (-12) = -15

  • -4 + (-9) = -13

  • -6 + (-6) = -12

By comparing these sums with the answer choices, we find that 15 and 12 are possible values of 'b'. If a single correct option was sought from the provided choices, the answers would be C. 12 and D. 15, as these values match one of the calculated sums. The factored form for b = 12 would be (x + 6)(x + 6), and for b = 15 it would be (x + 3)(x + 12).

Conclusion: Mastering Factoring through Practice

These practice questions illustrate the process of finding the value of 'b' for which a quadratic expression can be factored. By systematically identifying factor pairs, calculating their sums, and comparing them with the answer choices, you can confidently solve these types of problems. Consistent practice is key to mastering factoring techniques and building a strong foundation in algebra. Remember to always verify your answer by factoring the expression and expanding it to ensure it matches the original quadratic expression.