Finding P(2) For Quadratic Functions F(x) And G(x)
#Introduction
In the realm of mathematics, quadratic functions hold a significant place, particularly within algebra and calculus. Understanding their properties, behavior, and relationships is crucial for solving a myriad of problems. In this article, we delve into a problem involving two quadratic functions, f(x) and g(x), and their difference, p(x). Our main objective is to find the value of p(2) given specific conditions related to the roots and values of p(x). We aim to provide a step-by-step analysis of the problem, shedding light on the underlying mathematical concepts and techniques used to arrive at the solution. This exploration not only serves as an exercise in problem-solving but also as a means to deepen our understanding of quadratic functions and their applications. By focusing on the core elements of the problem, we can unravel the intricacies and gain valuable insights into the behavior of these fundamental mathematical entities. In the following sections, we will dissect the problem statement, formulate a strategic approach, and execute the necessary calculations to determine the value of p(2). Understanding quadratic functions is the key, and we will emphasize how leveraging the properties of these functions leads us to a precise and accurate solution.
Problem Statement and Initial Analysis
To accurately determine the value of p(2), we first need to meticulously analyze the given information. Let's break down the problem statement: We have two distinct quadratic functions, f(x) = ax^2 + bx + c and g(x) = a_1x^2 + bx + c_1, where a ≠a_1. This distinction is crucial because it ensures that the difference between these quadratic functions remains a quadratic function. The function p(x) is defined as the difference between f(x) and g(x), i.e., p(x) = f(x) - g(x). The problem states that p(x) = 0 only for x = -1. This piece of information suggests that -1 is the unique root of the quadratic equation p(x) = 0, indicating that p(x) has a repeated root at x = -1. Furthermore, we are given that p(-2) = 2. This provides a specific point on the quadratic function p(x), which will be instrumental in determining the coefficients of p(x). The ultimate goal is to find the value of p(2). To achieve this, we need to first establish the explicit form of the quadratic function p(x) using the provided conditions. The fact that p(x) has a unique root at x = -1 implies that it can be written in the form p(x) = k(x + 1)^2, where k is a constant. This form captures the essence of a quadratic function with a repeated root. We can then use the condition p(-2) = 2 to solve for the constant k. Once we determine k, we will have the complete expression for p(x), allowing us to easily calculate p(2). This structured approach ensures that we systematically utilize all the given information to arrive at the correct solution. The initial analysis highlights the importance of recognizing the properties of quadratic functions and how specific conditions, such as the presence of a unique root and a given point, can be leveraged to determine the function's parameters. In the following sections, we will execute these steps to find the value of p(2).
Determining the Form of p(x)
The key to solving this problem lies in understanding how to express p(x) using the information provided. Since p(x) = f(x) - g(x), we can write it as:
p(x) = (ax^2 + bx + c) - (a_1x^2 + bx + c_1)
Simplifying this expression, we get:
p(x) = (a - a_1)x^2 + (b - b)x + (c - c_1)
p(x) = (a - a_1)x^2 + (c - c_1)
Let's denote A = a - a_1 and C = c - c_1. Then, p(x) can be written as:
p(x) = Ax^2 + C
Now, we are given that p(x) = 0 only for x = -1. This means that -1 is the only root of the equation Ax^2 + C = 0. In other words, the quadratic equation has a repeated root at x = -1. This implies that p(x) can be written in the form:
p(x) = k(x + 1)^2, where k is a constant.
Expanding this, we get:
p(x) = k(x^2 + 2x + 1)
p(x) = kx^2 + 2kx + k
This form is crucial because it captures the essence of a quadratic function with a unique root. The unique root condition significantly simplifies the problem, allowing us to express p(x) in terms of a single constant k. The next step is to determine the value of k using the additional information provided, namely that p(-2) = 2. This condition will provide the necessary constraint to solve for the unknown constant. The ability to express p(x) in this form demonstrates a deep understanding of quadratic functions and their properties. By leveraging the fact that p(x) has a repeated root, we have transformed the problem into a more manageable form. This strategic approach is a testament to the power of mathematical reasoning and problem-solving skills. In the subsequent sections, we will use the condition p(-2) = 2 to find the value of k and ultimately determine the value of p(2).
Solving for the Constant k
Having established that p(x) = k(x + 1)^2, our next crucial step is to determine the value of the constant k. We are given the condition that p(-2) = 2. This piece of information provides us with a specific point on the graph of p(x), which we can use to solve for k. Substituting x = -2 into the expression for p(x), we get:
p(-2) = k((-2) + 1)^2
2 = k(-1)^2
2 = k(1)
k = 2
Thus, we have found that k = 2. This value is critical as it completes the definition of the quadratic function p(x). Now we can write p(x) explicitly as:
p(x) = 2(x + 1)^2
This expression fully defines the function p(x), allowing us to calculate its value for any given x. The process of solving for k demonstrates the importance of utilizing all available information in a problem. The condition p(-2) = 2 served as the key to unlocking the value of k, which in turn allowed us to fully characterize p(x). The ability to manipulate equations and solve for unknowns is a fundamental skill in mathematics, and this step exemplifies its importance. With the expression for p(x) now fully determined, we are well-positioned to calculate p(2), which is the final goal of the problem. The next section will focus on this calculation, bringing us to the ultimate solution. The strategic approach of first finding the general form of p(x) and then solving for the constant k has proven to be effective. This methodical approach is a hallmark of strong problem-solving skills in mathematics, ensuring accuracy and clarity in the solution process.
Calculating p(2)
With the explicit form of p(x) determined as p(x) = 2(x + 1)^2, the final step in solving the problem is to calculate the value of p(2). This involves substituting x = 2 into the expression for 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 result completes the solution to the problem. The calculation itself is straightforward, but it is the culmination of all the preceding steps. The initial analysis of the problem, the determination of the form of p(x), and the solving for the constant k all led to this final calculation. The fact that p(2) = 18 is the result of a logical and methodical approach underscores the importance of careful problem-solving strategies in mathematics. This final step not only provides the answer but also reinforces the understanding of how quadratic functions behave and how specific conditions can be used to determine their values. The process of calculating p(2) highlights the elegance and precision of mathematics. By systematically applying the given information and leveraging the properties of quadratic functions, we have arrived at a definitive answer. This demonstrates the power of mathematical reasoning and the satisfaction that comes from solving a problem through careful analysis and execution. In the conclusion, we will summarize the steps taken and the key concepts utilized to solve this problem, further solidifying our understanding of quadratic functions.
Conclusion
In summary, we have successfully determined the value of p(2) to be 18 by employing a structured and analytical approach. The problem involved two quadratic functions, f(x) and g(x), and their difference, p(x). We were given the conditions that p(x) = 0 only for x = -1 and p(-2) = 2. Using these conditions, we methodically worked through the problem to arrive at the solution.
First, we expressed p(x) as the difference between f(x) and g(x), which led us to the general form p(x) = (a - a_1)x^2 + (c - c_1). Recognizing that p(x) has a repeated root at x = -1, we then expressed p(x) in the form p(x) = k(x + 1)^2, where k is a constant. This step was crucial as it simplified the problem by allowing us to work with a single unknown constant.
Next, we used the condition p(-2) = 2 to solve for k. Substituting x = -2 into the equation p(x) = k(x + 1)^2, we found that k = 2. This gave us the explicit form of p(x) as p(x) = 2(x + 1)^2.
Finally, we calculated p(2) by substituting x = 2 into the expression for p(x). This yielded p(2) = 2(2 + 1)^2 = 18, which is the solution to the problem.
This problem highlights several key concepts in mathematics, including the properties of quadratic functions, the significance of roots, and the importance of using given conditions to solve for unknowns. The ability to manipulate equations, recognize patterns, and apply logical reasoning are essential skills demonstrated throughout the solution process. Furthermore, the problem showcases the power of a systematic approach to problem-solving. By breaking the problem down into smaller, manageable steps, we were able to effectively utilize the given information and arrive at the correct answer.
In conclusion, the value of p(2) is 18, and the solution process has provided valuable insights into the behavior and properties of quadratic functions. This exercise serves as a reminder of the beauty and precision of mathematics and the satisfaction that comes from solving complex problems through careful analysis and execution.
