Solving for x in the Function f(x) x2 - 3x: A Comprehensive Guide
Hello and welcome to this detailed guide on how to solve for x in the function f(x) x2 - 3x. This article is designed to help students, educators, and anyone with an interest in algebra and calculus understand the process step-by-step. Let's start by breaking down the function and then we'll dive into the specific example of solving for x -8 in this function.
Understanding the Function f(x) x2 - 3x
The function f(x) x2 - 3x is a quadratic function, which is a type of polynomial function. Quadratic functions are of the form f(x) ax2 bx c, where a, b, and c are constants and a ≠ 0. In our case, a 1, b -3, and c 0.
The primary purpose of solving for x in a quadratic function is to find the points where the function intersects the x-axis. This is also known as finding the roots or zeros of the function. To do this, we set the function equal to zero and solve for x:
f(x) x^2 - 3x 0
Let's solve this equation step-by-step:
x^2 - 3x 0
x(x - 3) 0
x 0 or x - 3 0 Therefore, x 0 or x 3
These are the roots of the function f(x) x2 - 3x.
Solving for x -8 in the Function f(x) x2 - 3x
Now, we will demonstrate how to find the value of the function f(-8) using the given function f(x) x2 - 3x.
First, we replace x with -8 in the function:
f(-8) (-8)^2 - 3(-8)
Next, we carry out the operations:
(-8)^2 64
-3(-8) 24
64 24 88
Therefore, the value of the function f(-8) is 88.
Additional Tips and Tricks for Solving Quadratic Functions
Below are some additional tips and tricks for solving quadratic functions like f(x) x2 - 3x:
Factoring: As shown in the earlier example, factoring the quadratic equation can help find the roots quickly. This method is particularly effective when the quadratic can be easily factored.
Completing the Square: This method involves converting the quadratic equation into a perfect square trinomial. It is particularly useful in graphing and understanding the vertex form of the quadratic function.
The Quadratic Formula: The quadratic formula, x [-b ± √(b^2 - 4ac)] / 2a, can be used to solve any quadratic equation, even if factoring or completing the square is not straightforward. This formula is essential to know as it works for all quadratic equations.
Conclusion
Understanding and solving quadratic functions is a fundamental skill in algebra and calculus. With this guide, you now have a clear understanding of how to solve for x in the function f(x) x2 - 3x, as well as additional techniques to tackle similar problems. Practice these methods and you will become proficient in solving quadratic functions.
If you need more resources and examples, refer to your textbooks, online tutorials, or mathematics forums. Remember, the more you practice, the better you will become.