Finding P(2) For Quadratic Functions F(x) And G(x)
In the fascinating realm of mathematics, quadratic functions hold a special place. They are characterized by their parabolic curves and the rich interplay between their coefficients and roots. This article delves into a problem involving two quadratic functions, f(x) and g(x), and their difference, p(x). By leveraging the properties of quadratic equations and the given conditions, we aim to determine the value of p(2). This exploration will not only showcase the elegance of quadratic functions but also highlight the problem-solving techniques applicable in various mathematical contexts.
Before diving into the specifics of the problem, let's briefly recap the fundamental aspects of quadratic functions. A quadratic function is generally expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants, and a ≠ 0. The roots of a quadratic equation are the values of x for which f(x) = 0. These roots can be found using the quadratic formula or by factoring the quadratic expression. The discriminant, Δ = b² - 4ac, plays a crucial role in determining the nature of the roots. If Δ > 0, the equation has two distinct real roots; if Δ = 0, it has one real root (a repeated root); and if Δ < 0, it has no real roots.
The problem at hand involves two distinct quadratic functions, f(x) and g(x), with leading coefficients a and a₁, respectively, where a ≠ a₁ and both are non-zero. The difference between these functions, p(x) = f(x) - g(x), is also a quadratic function. We are given that p(x) = 0 only for x = -1, indicating that -1 is a repeated root of p(x). Additionally, we know that p(-2) = 2. Our objective is to find the value of p(2). This problem elegantly combines the concepts of quadratic functions, roots, and function evaluation, requiring a careful and systematic approach to solve.
Let's formally state the problem to ensure a clear understanding of the given information and the objective. We are given two quadratic functions:
f(x) = ax² + bx + c g(x) = a₁x² + b₁x + c₁
where a ≠ a₁ and both are non-zero. We define a new function p(x) as the difference between f(x) and g(x):
p(x) = f(x) - g(x)
We are provided with two key pieces of information:
- p(x) = 0 only for x = -1
- p(-2) = 2
The core task is to determine the value of p(2). This problem hinges on our ability to effectively utilize the given information about the roots and function values of p(x) to deduce its explicit form and subsequently evaluate it at x = 2. The condition that p(x) = 0 only for x = -1 is particularly crucial, as it implies that p(x) has a repeated root at x = -1. This allows us to express p(x) in a specific form that simplifies the subsequent calculations. Furthermore, the information p(-2) = 2 provides a concrete data point that we can use to determine the remaining unknown coefficient in the expression for p(x).
To tackle this problem effectively, we need to recall the properties of quadratic functions and their roots. Specifically, if a quadratic function has a repeated root at x = r, then it can be expressed in the form k(x - r)², where k is a constant. This representation is a direct consequence of the fact that a quadratic equation with a repeated root has the form (x - r)² = 0. In our case, since p(x) = 0 only for x = -1, we can express p(x) as k(x + 1)², where k is a constant that we need to determine. The condition p(-2) = 2 will then allow us to find the value of k. Once we have the explicit form of p(x), evaluating p(2) becomes a straightforward substitution.
Now, let's embark on the solution process, breaking it down into manageable steps to ensure clarity and accuracy.
Step 1: Expressing p(x) in terms of its repeated root
As established in the problem statement, p(x) = f(x) - g(x). Since both f(x) and g(x) are quadratic functions, their difference, p(x), will also be a quadratic function. Given that p(x) = 0 only for x = -1, this implies that p(x) has a repeated root at x = -1. Therefore, we can express p(x) in the form:
p(x) = k(x + 1)²
where k is a constant that we need to determine. This form captures the essence of the repeated root condition, as the squared term (x + 1)² ensures that p(x) is zero only when x = -1. The constant k scales the quadratic expression and determines the overall shape of the parabola.
Step 2: Determining the value of k using the condition p(-2) = 2
To find the value of k, we utilize the given condition p(-2) = 2. Substituting x = -2 into the expression for p(x), we get:
2 = k(-2 + 1)² 2 = k(-1)² 2 = k
Thus, we have found that k = 2. This step is crucial as it allows us to determine the specific quadratic function p(x) that satisfies the given conditions. The value of k scales the quadratic term and dictates the vertical stretch or compression of the parabola.
Step 3: Writing the explicit form of p(x)
Now that we have found k = 2, we can write the explicit form of p(x):
p(x) = 2(x + 1)²
This equation fully defines the quadratic function p(x). It encapsulates the information about the repeated root at x = -1 and the scaling factor determined by the condition p(-2) = 2. This explicit form is the key to evaluating p(2) and answering the problem.
Step 4: Evaluating p(2)
Finally, to find p(2), we substitute x = 2 into the explicit form of p(x):
p(2) = 2(2 + 1)² p(2) = 2(3)² p(2) = 2(9) p(2) = 18
Therefore, the value of p(2) is 18. This is the final answer to the problem, obtained by systematically utilizing the given information and the properties of quadratic functions.
Based on our step-by-step solution, the value of p(2) is 18. This corresponds to option (1) in the given choices. This answer is the culmination of our efforts, showcasing the power of mathematical reasoning and problem-solving techniques.
In conclusion, we have successfully determined the value of p(2) by leveraging the properties of quadratic functions and the given conditions. The problem beautifully illustrates how the roots and function values of a quadratic function can be used to deduce its explicit form. The key insight was recognizing that the condition p(x) = 0 only for x = -1 implies a repeated root, allowing us to express p(x) in the form k(x + 1)². The subsequent use of the condition p(-2) = 2 enabled us to find the value of k and, ultimately, p(2).
This problem not only reinforces our understanding of quadratic functions but also highlights the importance of systematic problem-solving in mathematics. By breaking down the problem into smaller, manageable steps, we were able to navigate through the complexities and arrive at the correct solution. The ability to translate given information into mathematical expressions and equations is a crucial skill in various mathematical disciplines.
The techniques employed in this problem can be extended to other scenarios involving quadratic functions and their properties. For instance, similar approaches can be used to find the equation of a parabola given its vertex and a point on the curve, or to determine the roots of a quadratic equation given certain conditions. The core principles of utilizing the relationship between roots, coefficients, and function values remain applicable in a wide range of problems.
Furthermore, this problem serves as a reminder of the interconnectedness of mathematical concepts. The solution required a combination of knowledge about quadratic functions, roots, and function evaluation. This holistic approach to problem-solving is essential for success in mathematics and related fields. By mastering the fundamental concepts and developing strong problem-solving skills, we can unlock the beauty and power of mathematics.
In summary, the problem of finding p(2) given the specific conditions has been a rewarding exercise in mathematical reasoning. The solution not only provides a concrete answer but also reinforces our understanding of quadratic functions and their properties. The techniques employed and the insights gained will undoubtedly be valuable in tackling future mathematical challenges.
