Factoring Quadratics A Deep Dive Into Integer Constants And Expressions

In the realm of algebra, quadratic expressions often present a fascinating puzzle. These expressions, characterized by their highest power of two, can be manipulated and rewritten in various forms, revealing hidden relationships between their coefficients and constants. One common technique is factoring, where a quadratic expression is decomposed into a product of two linear expressions. This article delves into the intricacies of factoring a specific quadratic expression and explores the conditions under which certain resulting quantities must be integers. We will analyze the expression $4x^2 + bx - 45$, where $b$ is a constant, and its factored form $(hx + k)(x + j)$, where $h$, $k$, and $j$ are integer constants. Our focus will be on determining which of the given options, involving ratios of these constants, must be an integer.

Understanding the Problem

The problem presents a quadratic expression, $4x^2 + bx - 45$, and its factored form, $(hx + k)(x + j)$. The key information is that $b$, $h$, $k$, and $j$ are all integer constants. This constraint is crucial because it limits the possible values and relationships between these constants. Our goal is to identify which of the following expressions must result in an integer: (A) $\frac{b}{h}$, (B) $\frac{b}{k}$, (C) $\frac{45}{h}$, (D) $\frac{45}{k}$. To solve this, we need to expand the factored form, compare coefficients with the original quadratic expression, and analyze the resulting equations.

Expanding the Factored Form

Let's begin by expanding the factored form $(hx + k)(x + j)$:

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

Now, we have an expanded quadratic expression in terms of $h$, $j$, and $k$. To relate this back to the original expression, we'll compare the coefficients of corresponding terms.

Comparing Coefficients

Comparing the expanded form $hx^2 + (hj + k)x + kj$ with the original quadratic expression $4x^2 + bx - 45$, we can establish the following equations:

1. Coefficient of $x^2$: $h = 4$
2. Coefficient of $x$: $b = hj + k$
3. Constant term: $kj = -45$

These three equations form the foundation for our analysis. The first equation tells us that $h$ must be equal to 4. The second equation expresses $b$ in terms of $h$, $j$, and $k$. The third equation establishes a relationship between $k$ and $j$.

Analyzing the Options

Now, let's examine each of the given options to determine which must be an integer.

(A) $\frac{b}{h}$

We know that $b = hj + k$ and $h = 4$. Therefore, $\frac{b}{h} = \frac{hj + k}{h} = \frac{4j + k}{4} = j + \frac{k}{4}$. For $\frac{b}{h}$ to be an integer, $\frac{k}{4}$ must be an integer. However, we only know that $kj = -45$, and $k$ and $j$ are integers. This does not guarantee that $k$ is divisible by 4. For instance, if $k = 15$ and $j = -3$, then $kj = -45$, but $\frac{k}{4} = \frac{15}{4}$ is not an integer. Therefore, option (A) is not necessarily an integer.

(B) $\frac{b}{k}$

Using $b = hj + k$, we have $\frac{b}{k} = \frac{hj + k}{k} = \frac{hj}{k} + 1$. For $\frac{b}{k}$ to be an integer, $\frac{hj}{k}$ must be an integer. Since $h = 4$, we have $\frac{4j}{k}$. We know $kj = -45$, so $j = -\frac{45}{k}$. Substituting this into $\frac{4j}{k}$, we get $\frac{4(-\frac{45}{k})}{k} = -\frac{180}{k^2}$. This expression is not necessarily an integer. For example, if $k = 1$, then $j = -45$, and $\frac{b}{k} = \frac{4(-45) + 1}{1} = -179$, which is an integer. However, if $k=2$, then it won't be an integer. Therefore, option (B) is not necessarily an integer.

(C) $\frac{45}{h}$

Since $h = 4$, we have $\frac{45}{h} = \frac{45}{4}$. This is clearly not an integer. Thus, option (C) is not an integer.

(D) $\frac{45}{k}$

From the equation $kj = -45$, we can express $j$ as $j = -\frac{45}{k}$. Since $j$ is an integer, it follows that $\frac{45}{k}$ must be an integer (or its negative must be an integer, which implies the expression's absolute value must be an integer). Therefore, $k$ must be a factor of 45. The factors of 45 are $\pm 1$, $\pm 3$, $\pm 5$, $\pm 9$, $\pm 15$, and $\pm 45$. For each of these values of $k$, $j$ will be an integer. Thus, option (D) must be an integer.

Conclusion

Through a careful analysis of the given quadratic expression, its factored form, and the constraints on the constants, we have determined that only option (D), $\frac{45}{k}$, must be an integer. This conclusion is derived from the relationship $kj = -45$, which directly implies that $k$ must be a factor of 45, and therefore, $\frac{45}{k}$ will always be an integer. This exercise demonstrates the power of algebraic manipulation and coefficient comparison in solving problems involving quadratic expressions and their factors.

The Importance of Integer Constraints

The problem's condition that $h$, $k$, and $j$ are integers is absolutely crucial. Without this constraint, the possibilities for the values of these constants become vastly broader, and none of the options would necessarily result in an integer. The integer constraint narrows down the possible factor pairs of -45, allowing us to make definitive conclusions about the relationships between the constants. This highlights the significance of paying close attention to the given constraints in mathematical problems, as they often provide key insights and guide the solution process.

Factoring Quadratics in General

The techniques used in this problem, such as expanding factored forms and comparing coefficients, are fundamental to factoring quadratics in general. Factoring a quadratic expression involves finding two binomials whose product equals the given quadratic. The process often involves identifying factor pairs of the constant term and testing different combinations to find the correct middle term. When the leading coefficient (the coefficient of the $x^2$ term) is not 1, as in this case, the factoring process can become more complex, requiring careful consideration of the factors of both the leading coefficient and the constant term.

Real-World Applications of Quadratic Expressions

Quadratic expressions and their factored forms have numerous applications in various fields, including physics, engineering, and economics. For example, the trajectory of a projectile can be modeled using a quadratic equation, and the roots of the equation (the values of $x$ that make the expression equal to zero) represent the points where the projectile hits the ground. In engineering, quadratic equations are used to design bridges, buildings, and other structures. In economics, they can model supply and demand curves and help determine equilibrium prices. The ability to manipulate and factor quadratic expressions is therefore a valuable skill in many disciplines.

Summary of Key Concepts

• Factoring Quadratics: Decomposing a quadratic expression into a product of two linear expressions.
• Coefficient Comparison: Equating coefficients of corresponding terms in two equivalent polynomial expressions.
• Integer Constraints: Conditions that restrict variables to integer values, which often simplify problem-solving.
• Factors and Divisibility: Understanding the relationship between factors of a number and divisibility.
• Real-World Applications: Recognizing the diverse applications of quadratic expressions in various fields.

By mastering these concepts and techniques, students can confidently tackle a wide range of problems involving quadratic expressions and their factored forms.