The Expression $4x^2 + Bx - 45$, Where $b$ Is A Constant, Can Be Rewritten As $(hx + K)(x + J)$, Where $ H , K , H, K, H , K , [/tex] And $j$ Are Integer Constants. Which Of The Following Must Be An Integer?A)

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Introduction

In algebra, factorization is a crucial concept that helps us simplify complex expressions and solve equations. The expression $4x^2 + bx - 45$ is a quadratic expression that can be rewritten in the form $(hx + k)(x + j)$, where $h, k,$ and $j$ are integer constants. In this article, we will explore the factorization of the given expression and determine which of the following must be an integer.

Understanding the Factorization

To factorize the expression $4x^2 + bx - 45$, we need to find two binomials whose product is equal to the given expression. The general form of the factorization is $(hx + k)(x + j)$. When we multiply these two binomials, we get:

$$(hx + k)(x + j) = hx^2 + hjx + kx + kj$$

Expanding the right-hand side, we get:

$$hx^2 + (hj + k)x + kj$$

Comparing the coefficients of the terms on both sides, we can see that:

• The coefficient of $x^2$ is $h$.
• The coefficient of $x$ is $hj + k$.
• The constant term is $kj$.

Equating Coefficients

Now, let's equate the coefficients of the terms in the given expression $4x^2 + bx - 45$ with the coefficients of the terms in the factorized form $(hx + k)(x + j)$.

• The coefficient of $x^2$ in the given expression is $4$, so we have $h = 4$.
• The coefficient of $x$ in the given expression is $b$, so we have $hj + k = b$.
• The constant term in the given expression is $-45$, so we have $kj = -45$.

Solving for $j$ and $k$

Since $h = 4$, we can substitute this value into the equation $kj = -45$ to get:

$$4j = -45$$

Dividing both sides by $4$, we get:

$$j = -\frac{45}{4}$$

However, we are told that $j$ is an integer. Therefore, we must have $j = -9$ or $j = 5$, since these are the only integer factors of $-45$ that divide $4$.

Determining the Value of $k$

Now that we have found the possible values of $j$, we can determine the corresponding values of $k$. Since $kj = -45$, we have:

• If $j = -9$, then $k = 5$.
• If $j = 5$, then $k = -9$.

Conclusion

In conclusion, the expression $4x^2 + bx - 45$ can be rewritten as $(hx + k)(x + j)$, where $h, k,$ and $j$ are integer constants. We have found that $h = 4$ and $j = -9$ or $j = 5$. Therefore, the value of $k$ must be $5$ or $-9$, respectively.

Answer

The correct answer is that $k$ must be an integer.

Final Thoughts

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Introduction

In our previous article, we explored the factorization of the expression $4x^2 + bx - 45$ and determined which of the following must be an integer. In this article, we will answer some frequently asked questions related to the factorization of the given expression.

Q: What is the general form of the factorization of the expression $4x^2 + bx - 45$?

A: The general form of the factorization is $(hx + k)(x + j)$, where $h, k,$ and $j$ are integer constants.

Q: How do we find the values of $h, k,$ and $j$?

A: To find the values of $h, k,$ and $j$, we need to equate the coefficients of the terms in the given expression $4x^2 + bx - 45$ with the coefficients of the terms in the factorized form $(hx + k)(x + j)$. We have:

• $h = 4$
• $hj + k = b$
• $kj = -45$

Q: What are the possible values of $j$?

A: Since $h = 4$, we can substitute this value into the equation $kj = -45$ to get:

$$4j = -45$$

Dividing both sides by $4$, we get:

$$j = -\frac{45}{4}$$

However, we are told that $j$ is an integer. Therefore, we must have $j = -9$ or $j = 5$, since these are the only integer factors of $-45$ that divide $4$.

Q: What are the corresponding values of $k$?

A: Since $kj = -45$, we have:

• If $j = -9$, then $k = 5$.
• If $j = 5$, then $k = -9$.

Q: Why must $k$ be an integer?

A: Since $kj = -45$, and $j$ is an integer, $k$ must also be an integer.

Q: Can we find other values of $h, k,$ and $j$?

A: No, we have found all possible values of $h, k,$ and $j$ that satisfy the given conditions.

Q: How can we use this factorization to solve equations?

A: We can use this factorization to solve equations by multiplying the two binomials and equating the resulting expression with the given equation.

Q: What are some real-world applications of this factorization?

A: This factorization has many real-world applications, such as solving quadratic equations that arise in physics, engineering, and economics.

Conclusion

In this article, we have answered some frequently asked questions related to the factorization of the expression $4x^2 + bx - 45$. We have found that $k$ must be an integer, and we have also found the possible values of $j$ and $k$. This problem is a great example of how factorization can be used to simplify complex expressions and solve equations.

Final Thoughts

In this article, we have explored the factorization of the expression $4x^2 + bx - 45$ and answered some frequently asked questions related to the factorization. We hope that this article has been helpful in understanding the concept of factorization and its applications.