Determine Value Of X For Undefined Quotient (2x^2 + 3x - 4) / (x + 4)

JU07/09/2025 08, 2025 by THE IDEN 70 views

$For What Value of $x$ Would the Quotient $\left(2 x^2+3 x-4\right) \div(x+4)$ Not Make Sense?$

In the realm of mathematics, understanding the conditions under which expressions are defined is crucial. When dealing with quotients, or fractions, a key principle is that division by zero is undefined. This article delves into the question of determining the value of $x$ for which the quotient $\left(2 x^2+3 x-4\right) \div(x+4)$ becomes undefined. We will explore the underlying mathematical principles, walk through the solution process, and discuss the broader implications of this concept.

Identifying the Problematic Value of $x$

To determine for what value of $x$ the quotient $\left(2 x^2+3 x-4\right) \div(x+4)$ does not make sense, we need to identify when the denominator, $(x+4)$, equals zero. In mathematical terms, a rational expression (a fraction with polynomials) is undefined when its denominator is zero. This is because division by zero is an undefined operation in mathematics. It leads to contradictions and inconsistencies within the mathematical system. Understanding this fundamental principle is essential for working with algebraic expressions and equations. When we divide by zero, we violate the basic axioms of arithmetic, leading to nonsensical results. This is why it's crucial to identify and exclude values that make the denominator zero.

To find the value of $x$ that makes the denominator zero, we set the denominator equal to zero and solve for $x$: $x + 4 = 0$. This simple equation allows us to pinpoint the exact value of $x$ that causes the expression to be undefined. By isolating $x$, we can determine the specific input that results in division by zero, which is the core of our problem. Recognizing this step is crucial for solving similar problems involving rational expressions. It's a standard technique used to identify points of discontinuity or undefined behavior in functions.

Subtracting 4 from both sides of the equation, we get: $x = -4$. This result tells us that when $x$ is equal to -4, the denominator of the quotient becomes zero. Therefore, the quotient $\left(2 x^2+3 x-4\right) \div(x+4)$ is undefined when $x = -4$. This is because substituting -4 for $x$ in the denominator results in $(-4) + 4 = 0$, leading to division by zero. This value, $x = -4$, is the critical point where the expression ceases to be mathematically meaningful. It's important to note that the numerator of the expression does not play a role in determining when the expression is undefined. The issue arises solely from the denominator being zero.

Why Division by Zero is Undefined

To further understand why the quotient doesn't make sense when $x = -4$, let's delve into the mathematical reason behind the undefined nature of division by zero. Division can be thought of as the inverse operation of multiplication. For example, $12 \div 3 = 4$ because $4 \times 3 = 12$. Similarly, $a \div b = c$ if and only if $c \times b = a$. Now, let's consider what happens when we divide by zero. Suppose we have an expression like $k \div 0$, where $k$ is any non-zero number. If this division were to yield a result, let's call it $c$, then we would have $k \div 0 = c$. This would imply that $c \times 0 = k$. However, any number multiplied by zero is zero, so we would have $0 = k$. This is a contradiction since we assumed $k$ was a non-zero number. This contradiction illustrates why division by zero cannot be defined in a consistent manner within the mathematical system.

If we consider the case where $k = 0$, we encounter a different problem. Suppose $0 \div 0 = c$. This would imply that $c \times 0 = 0$. But this equation is true for any value of $c$, meaning that $0 \div 0$ could be any number. This lack of a unique solution makes division by zero undefined. The result would be ambiguous and not mathematically useful. Therefore, to maintain the consistency and integrity of mathematical operations, division by zero is explicitly excluded. This principle is a cornerstone of arithmetic and algebra, ensuring that our calculations and equations remain logically sound.

In the context of the given problem, when $x = -4$, the denominator $(x + 4)$ becomes zero. This leads to an expression of the form $\frac{2(-4)^2 + 3(-4) - 4}{0}$, which involves division by zero. As we have discussed, this operation is undefined, making the entire quotient undefined for this value of $x$. This understanding is critical for identifying restrictions on the domain of rational functions and for avoiding errors in algebraic manipulations. The value $x = -4$ represents a point of discontinuity in the function represented by the quotient.

Implications and Applications

Understanding when a quotient is undefined has significant implications in various areas of mathematics. In algebra, it helps in identifying the domain of rational functions. A rational function is a function that can be expressed as the quotient of two polynomials. The domain of a rational function is the set of all real numbers except those that make the denominator zero. For the given quotient, $f(x) = \frac{2x^2 + 3x - 4}{x + 4}$, the domain is all real numbers except $x = -4$. This means that the function is defined for any value of $x$ other than -4.

In calculus, understanding undefined quotients is crucial when dealing with limits and continuity. A limit is the value that a function approaches as the input approaches some value. If a function is undefined at a particular point, we can still investigate its behavior as the input gets arbitrarily close to that point. However, the function's value at that point remains undefined. This concept is fundamental to the study of calculus, allowing us to analyze the behavior of functions near points of discontinuity. For instance, we can explore the limit of the given quotient as $x$ approaches -4, even though the function is undefined at $x = -4$.

Furthermore, in real-world applications, understanding undefined quotients can help in avoiding errors and misinterpretations. For example, in physics, certain formulas may involve quotients where the denominator represents a physical quantity. If that quantity can become zero under certain conditions, it's crucial to recognize that the formula may not be valid under those conditions. Similarly, in engineering and economics, models often involve rational expressions, and understanding their domains is essential for making accurate predictions and decisions. Ignoring the possibility of division by zero can lead to erroneous conclusions and potentially dangerous outcomes.

Conclusion

In summary, the quotient $\left(2 x^2+3 x-4\right) \div(x+4)$ does not make sense when the denominator, $(x+4)$, is equal to zero. By solving the equation $x + 4 = 0$, we find that $x = -4$ is the value that makes the quotient undefined. This is because division by zero is an undefined operation in mathematics. This concept is crucial in various mathematical contexts, including algebra, calculus, and real-world applications. Understanding the domain of rational expressions and avoiding division by zero are essential skills for anyone working with mathematical models and equations. By recognizing these fundamental principles, we can ensure the accuracy and validity of our mathematical reasoning.

Therefore, the correct answer is C. -4.