Solving F(x) = 2x³ - 3x² A Comprehensive Guide

Hey guys! Let's dive into the world of functions and tackle the equation f(x) = 2x³ - 3x². This might seem a bit intimidating at first, but don't worry, we'll break it down step by step. Understanding how to solve cubic functions like this one is super important in various fields, from engineering to computer science. So, grab your thinking caps, and let's get started!

Understanding the Basics of Cubic Functions

Before we jump into solving our specific function, let's quickly recap what cubic functions are all about. A cubic function is basically a polynomial function where the highest power of the variable (in our case, 'x') is 3. The general form of a cubic function looks something like this: f(x) = ax³ + bx² + cx + d, where 'a', 'b', 'c', and 'd' are constants. In our example, f(x) = 2x³ - 3x², we have a = 2, b = -3, c = 0, and d = 0. Recognizing this form is the first step in understanding how to approach the problem. The solutions to f(x) = 0 are also known as the roots or zeros of the function, which are the x-values where the graph of the function intersects the x-axis. Finding these roots is often the primary goal when solving cubic functions.

When dealing with cubic functions, it's essential to grasp their behavior. Cubic functions can have up to three real roots, which means the graph of the function can cross the x-axis at most three times. However, they can also have one real root and two complex roots. The shape of a cubic function's graph is also distinctive, often resembling an elongated 'S' curve. The leading coefficient, 'a', plays a crucial role in determining the graph's end behavior. If 'a' is positive, the graph rises to the right and falls to the left. If 'a' is negative, the graph falls to the right and rises to the left. These fundamental concepts lay the groundwork for effectively solving cubic functions, allowing us to predict the nature and number of solutions. Understanding these basics is crucial for effectively solving our specific problem and similar problems in the future.

Methods to Solve f(x) = 2x³ - 3x²

Now, let's get down to business and explore the methods we can use to solve f(x) = 2x³ - 3x². There are a few different approaches we can take, but for this particular function, one method stands out as being the most straightforward: factoring. Factoring is a technique where we try to express the function as a product of simpler expressions. This is a powerful method because if we can factor the function into the form (x - r)(something else) = 0, then we know that x = r is a root of the function. For more complex cubic functions, we might need to resort to numerical methods or formulas, but for our case, factoring will do the trick.

Factoring

Factoring is our primary weapon for this equation. When we look at f(x) = 2x³ - 3x², we can immediately notice that both terms have 'x' in them. In fact, they both have x² as a common factor. This is our key to unlocking the solution! We can factor out x² from the expression: f(x) = x²(2x - 3). Now, we have expressed our cubic function as a product of two factors: x² and (2x - 3). This makes it much easier to find the roots. By setting each factor equal to zero, we can solve for x. This method is particularly effective when the cubic function has a simple structure, like ours, where common factors are readily apparent. Factoring not only simplifies the equation but also provides a clear pathway to finding the solutions, making it an essential technique in algebra.

Setting Factors to Zero

After factoring, the next step is to set each factor equal to zero. This is based on the principle that if the product of two or more factors is zero, then at least one of the factors must be zero. In our case, we have two factors: x² and (2x - 3). So, we set x² = 0 and (2x - 3) = 0. Solving x² = 0 is straightforward: taking the square root of both sides gives us x = 0. This means x = 0 is a root of our function, and it's actually a repeated root because of the square. Now, let's tackle (2x - 3) = 0. We can solve this linear equation by adding 3 to both sides, which gives us 2x = 3. Then, dividing both sides by 2, we get x = 3/2. So, we have found another root: x = 3/2. Setting factors to zero is a crucial step in solving factored equations, as it directly leads us to the values of x that make the function equal to zero, which are the roots we're after. This process is fundamental in algebra and is used extensively in solving various types of equations.

Solutions for f(x) = 2x³ - 3x²

Alright, let's recap what we've found. By factoring and setting the factors to zero, we've discovered the solutions for f(x) = 2x³ - 3x². We found that x = 0 is a repeated root, meaning it appears twice, and x = 3/2 is another root. So, the function f(x) = 2x³ - 3x² equals zero when x is 0 or 3/2. These are the points where the graph of the function intersects the x-axis. It's always a good idea to double-check our solutions, and we can do that by plugging them back into the original equation. If we plug in x = 0, we get f(0) = 2(0)³ - 3(0)² = 0, which is correct. If we plug in x = 3/2, we get f(3/2) = 2(3/2)³ - 3(3/2)² = 2(27/8) - 3(9/4) = 27/4 - 27/4 = 0, which also confirms our solution. Knowing the solutions to a function allows us to understand its behavior and graph, which is super useful in many applications.

Graphing f(x) = 2x³ - 3x²

Now that we've found the solutions, let's take a peek at what the graph of f(x) = 2x³ - 3x² looks like. Graphing a function gives us a visual representation of its behavior, and it can help us confirm our solutions and understand the function even better. To graph this cubic function, we can start by plotting the roots we found: x = 0 and x = 3/2 (which is 1.5). Since x = 0 is a repeated root, the graph will touch the x-axis at x = 0 but not cross it. This means the graph will change direction at that point. We also know that this is a cubic function with a positive leading coefficient (2), so the graph will rise to the right and fall to the left.

With the roots and the end behavior in mind, we can sketch a basic graph. The graph will start from the bottom-left, go up, touch the x-axis at x = 0, change direction and go down, then turn again and cross the x-axis at x = 3/2, and continue rising to the top-right. To get a more accurate graph, we could also find some additional points by plugging in other x-values and calculating the corresponding f(x) values. For example, we could plug in x = -1 and x = 2 to get a better sense of the curve. Graphing calculators or online tools like Desmos can also be super helpful for visualizing the graph accurately. The graph will visually confirm our solutions and give us a complete picture of the function's behavior. Visualizing the function helps solidify our understanding and makes it easier to apply this knowledge to real-world problems.

Applications of Cubic Functions

Okay, we've cracked the code for solving f(x) = 2x³ - 3x², but you might be wondering,