Find The Quadratic Function With Zeros At X=-2 And X=5

Finding the quadratic function with specific zeros is a fundamental concept in algebra. This article will explore how to determine the quadratic function that has zeros at x = -2 and x = 5. We will analyze the given options and methodically arrive at the correct solution. Understanding zeros of a function is crucial as it provides key insights into the function’s behavior and graph. Let's delve into the process step by step to ensure a clear understanding of the underlying principles.

Understanding Zeros of a Function

In mathematics, the zeros of a function, also known as roots or x-intercepts, are the values of x for which the function f(x) equals zero. In other words, they are the points where the graph of the function intersects the x-axis. For a quadratic function, which is a polynomial of degree two, these zeros are particularly significant. A quadratic function can have at most two real zeros, which can be found using various methods such as factoring, completing the square, or using the quadratic formula. Understanding how to find and interpret zeros is crucial in many areas of mathematics and its applications, including calculus, physics, and engineering.

The Significance of Zeros

Zeros of a function provide crucial information about the function’s behavior. They help in identifying the points where the function changes its sign (from positive to negative or vice versa). In the context of a quadratic function, the zeros determine the x-intercepts of the parabola, which is the graphical representation of the quadratic function. The zeros, along with the vertex (the highest or lowest point on the parabola), provide a comprehensive understanding of the quadratic function’s graph. Moreover, zeros are essential in solving equations and inequalities, as they represent the solutions to the equation f(x) = 0. In practical applications, such as in physics, zeros can represent equilibrium points or critical conditions in a system.

Methods to Find Zeros

There are several methods to find the zeros of a quadratic function. One of the most common methods is factoring. If the quadratic expression can be factored into two linear factors, the zeros can be easily found by setting each factor equal to zero and solving for x. Another method is completing the square, which involves transforming the quadratic equation into a perfect square trinomial. This method is particularly useful when factoring is not straightforward. The quadratic formula is a general method that can be used to find the zeros of any quadratic function. It is derived from the method of completing the square and provides a direct way to calculate the zeros using the coefficients of the quadratic equation. Graphing the function is another visual method, where the x-intercepts represent the zeros. Each of these methods offers a different approach, and the choice of method often depends on the specific form of the quadratic equation and personal preference.

Forming a Quadratic Function from Zeros

To form a quadratic function given its zeros, we utilize the fact that if x = a and x = b are zeros of a function, then (x - a) and (x - b) are factors of the function. This principle stems from the factor theorem, which states that a polynomial f(x) has a factor (x - a) if and only if f(a) = 0. In the case of a quadratic function, if we know the two zeros, we can construct the function by multiplying the corresponding factors. This approach is fundamental in algebra and is widely used in various mathematical contexts. Understanding this process allows for the creation of functions with specific characteristics, which is essential in modeling real-world phenomena.

Constructing Factors from Zeros

The initial step in forming a quadratic function from its zeros is to construct the factors. If x = -2 is a zero, then the factor is (x - (-2)), which simplifies to (x + 2). Similarly, if x = 5 is a zero, the factor is (x - 5). These factors represent the linear expressions that, when multiplied together, will give us the quadratic function. The rationale behind this is that when x equals either -2 or 5, one of the factors will become zero, making the entire product zero, thus satisfying the condition for a zero of the function. This method is a direct application of the factor theorem and is a crucial concept in polynomial algebra.

Multiplying the Factors

After identifying the factors, the next step is to multiply them together. In our case, we multiply (x + 2) and (x - 5). This multiplication involves distributing each term in the first factor across the terms in the second factor. Specifically, we multiply x by both x and -5, and then we multiply 2 by both x and -5. This process yields the terms x², -5x, 2x, and -10. Combining like terms, we get the quadratic expression x² - 3x - 10. This resulting expression is the quadratic function that has the given zeros. The multiplication of factors is a fundamental algebraic technique, and mastering it is essential for working with polynomials and other mathematical functions.

General Form of a Quadratic Function

The general form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. The function we obtained by multiplying the factors, x² - 3x - 10, is in this general form, with a = 1, b = -3, and c = -10. The coefficient a determines the direction in which the parabola opens (upward if a > 0 and downward if a < 0), while b and c affect the position and shape of the parabola. Understanding the general form of a quadratic function is crucial for identifying and analyzing quadratic functions and their properties. It allows for a standardized way to represent and compare different quadratic functions.

Analyzing the Given Options

Now, let's analyze the given options to determine which function has zeros at x = -2 and x = 5. We have the following options:

A. f(x) = x² + 2x - 10 B. f(x) = x² - 2x - 10 C. f(x) = x² + 3x - 10 D. f(x) = x² - 3x - 10

To find the correct function, we can either test each option by plugging in the given zeros or compare the functions to the one we derived from the zeros. We will use both methods to ensure clarity and accuracy. This analytical approach is essential in problem-solving and helps to develop a deeper understanding of the concepts involved. The process of analyzing options and verifying solutions is a critical skill in mathematics and other scientific disciplines.

Testing Each Option

One method to verify the zeros of each function is to substitute x = -2 and x = 5 into each function and check if the result is zero. This direct substitution method is a straightforward way to confirm whether a given value is a zero of the function. For option A, substituting x = -2 gives f(-2) = (-2)² + 2(-2) - 10 = 4 - 4 - 10 = -10, which is not zero. Similarly, for option B, substituting x = -2 gives f(-2) = (-2)² - 2(-2) - 10 = 4 + 4 - 10 = -2, which is also not zero. For option C, substituting x = -2 gives f(-2) = (-2)² + 3(-2) - 10 = 4 - 6 - 10 = -12, which is not zero either. However, for option D, substituting x = -2 gives f(-2) = (-2)² - 3(-2) - 10 = 4 + 6 - 10 = 0, and substituting x = 5 gives f(5) = (5)² - 3(5) - 10 = 25 - 15 - 10 = 0. Thus, option D satisfies the condition for both zeros.

Comparing to the Derived Function

Another method to identify the correct function is to compare the given options with the quadratic function we derived from the zeros x = -2 and x = 5, which is f(x) = x² - 3x - 10. By comparing the coefficients of each option with the coefficients of our derived function, we can quickly determine the matching function. Option A, f(x) = x² + 2x - 10, has different coefficients. Option B, f(x) = x² - 2x - 10, also has different coefficients. Option C, f(x) = x² + 3x - 10, has a different coefficient for the x term. However, option D, f(x) = x² - 3x - 10, matches our derived function exactly. This comparison method provides a direct and efficient way to identify the correct function. It reinforces the understanding of how the zeros of a function relate to its algebraic form.

Conclusion

In conclusion, the function with zeros at x = -2 and x = 5 is f(x) = x² - 3x - 10. This was determined by constructing the quadratic function from its zeros and verifying the solution by testing each option. Understanding the relationship between the zeros of a function and its factors is crucial in solving quadratic equations and analyzing their properties. This article has provided a step-by-step guide to identifying quadratic functions with specific zeros, emphasizing the importance of factoring, the quadratic formula, and analytical comparison. Mastering these concepts is essential for further studies in algebra and related fields.

By understanding the significance of zeros, constructing functions from given zeros, and analyzing options methodically, we can confidently solve problems involving quadratic functions. The ability to manipulate and interpret quadratic functions is a valuable skill in various mathematical and real-world applications. Therefore, a thorough understanding of these concepts is highly beneficial for students and professionals alike.

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Which quadratic function has roots (zeros) at x = -2 and x = 5?

Title

Find the Quadratic Function with Zeros at x=-2 and x=5