Finding Zeros Of F(x)=5x^2-x+10 Algebraically A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into the exciting world of finding the zeros of a function algebraically. Specifically, we'll be tackling the function f(x) = 5x² - x + 10. Don't worry if this looks intimidating; we'll break it down step by step, making it super easy to understand. So, grab your calculators (or your mental math muscles), and let's get started!

Understanding Zeros of a Function

Before we jump into the nitty-gritty, let's make sure we're all on the same page about what “zeros of a function” actually means. In simple terms, the zeros of a function are the x-values that make the function equal to zero. Graphically, these are the points where the function's graph intersects the x-axis. Finding these zeros is a fundamental concept in algebra and calculus, and it pops up in various applications, from physics to engineering to economics. Think of them as the 'sweet spots' of the function, where the output is precisely zero.

When we talk about finding the zeros algebraically, we mean solving for these x-values using mathematical equations and techniques, rather than relying solely on graphing or numerical methods. This approach gives us exact solutions, which can be crucial in many scenarios. For quadratic functions like the one we're dealing with today, there are a couple of primary methods we can use: factoring, completing the square, or the quadratic formula. We'll focus on the quadratic formula here, as it's a reliable method that works for all quadratic equations, even those that are difficult or impossible to factor.

So, why is finding zeros so important? Well, the zeros of a function can tell us a lot about its behavior. For instance, in the context of a projectile's trajectory, the zeros might represent the points where the projectile hits the ground. In a business model, they could represent the break-even points where revenue equals cost. Understanding these zeros allows us to analyze, predict, and optimize various real-world situations. Plus, mastering this concept is a key step in your mathematical journey, opening doors to more advanced topics and applications.

Applying the Quadratic Formula

The quadratic formula is our trusty tool for finding the zeros of any quadratic equation in the form ax² + bx + c = 0. It's a powerful formula that you'll want to have in your mathematical toolkit. The formula itself looks like this:

x = (-b ± √(b² - 4ac)) / (2a)

Where:

• a is the coefficient of the x² term
• b is the coefficient of the x term
• c is the constant term

Now, let's apply this formula to our function, f(x) = 5x² - x + 10. First, we need to identify our a, b, and c values. Comparing our function to the standard form, we can see that:

• a = 5
• b = -1
• c = 10

Great! We have our values. Now, it's just a matter of plugging them into the quadratic formula. Let's do it:

x = (-(-1) ± √((-1)² - 4 * 5 * 10)) / (2 * 5)

Take a moment to follow along and make sure you understand where each value is coming from. It's crucial to get this substitution step right, as a small mistake here can throw off the entire solution.

Next, we need to simplify the expression. Let's start with the stuff under the square root, which is often called the discriminant. The discriminant can actually tell us a lot about the nature of the zeros – whether they are real, imaginary, or repeated. But we'll get to that in more detail later. For now, let's just focus on simplifying:

x = (1 ± √(1 - 200)) / 10

x = (1 ± √(-199)) / 10

Notice that we have a negative number under the square root. This tells us something very important: the zeros of this function are not real numbers. They are complex numbers. This is a key takeaway, as it means the graph of the function will not intersect the x-axis. It floats either entirely above or entirely below the x-axis. Don't worry; we'll handle the complex numbers like pros!

Simplifying the Solution

Okay, we've arrived at the point where we have a negative number under the square root, √(-199). This means our solutions will involve imaginary numbers. Remember that the imaginary unit, denoted by 'i', is defined as the square root of -1 (i = √(-1)). So, we can rewrite √(-199) as √199 * √(-1) = √199 * i. This is where the magic of complex numbers comes into play!

Now, let's plug this back into our equation:

x = (1 ± √199 * i) / 10

This is our simplified solution! We can also write it in the standard form for complex numbers, which is a + bi, where 'a' is the real part and 'b' is the imaginary part. To do this, we simply separate the fraction:

x = 1/10 ± (√199 / 10)i

So, our two complex zeros are:

x₁ = 1/10 + (√199 / 10)i

x₂ = 1/10 - (√199 / 10)i

These are complex conjugate pairs, which is a common occurrence when dealing with quadratic equations that have a negative discriminant. Notice how the real part (1/10) is the same for both zeros, while the imaginary part has opposite signs. This is a characteristic of complex conjugate pairs.

To recap, we've successfully found the zeros of the function f(x) = 5x² - x + 10 using the quadratic formula. We encountered imaginary numbers along the way, which tells us that the function does not intersect the x-axis. We simplified our solution and expressed it in the standard form for complex numbers. Give yourselves a pat on the back – you've tackled a challenging problem like true mathematicians!

The Discriminant and Its Significance

Earlier, we briefly mentioned the discriminant, which is the part of the quadratic formula under the square root: b² - 4ac. This little expression holds a wealth of information about the nature of the zeros of a quadratic function. It's like a mathematical detective, giving us clues about the solutions even before we fully solve the equation.

Let's explore how the discriminant helps us understand the zeros:

• If b² - 4ac > 0: This means we have a positive number under the square root. In this case, we'll have two distinct real zeros. Graphically, this corresponds to the parabola intersecting the x-axis at two different points. These zeros represent two unique solutions to the quadratic equation, and they often have significant interpretations in real-world contexts.

• If b² - 4ac = 0: This means the expression under the square root is zero. In this scenario, the quadratic formula simplifies, and we get one real zero (which is sometimes referred to as a repeated or double root). Graphically, the parabola touches the x-axis at exactly one point, the vertex of the parabola. This situation represents a critical point where the function changes direction without crossing the x-axis.

• If b² - 4ac < 0: This is the situation we encountered in our example. A negative number under the square root means we have complex zeros. As we saw, these zeros involve the imaginary unit 'i'. Graphically, this means the parabola does not intersect the x-axis at all. It floats either entirely above or entirely below the x-axis. Complex zeros often arise in situations where we're dealing with oscillations, rotations, or other phenomena that don't have real-number solutions.

For our function, f(x) = 5x² - x + 10, we calculated the discriminant as (-1)² - 4 * 5 * 10 = 1 - 200 = -199. Since -199 is less than 0, we knew right away that we would have complex zeros. This insight saved us time and helped us anticipate the form of our solution.

Understanding the discriminant is a powerful tool in your mathematical arsenal. It allows you to quickly assess the nature of the zeros of a quadratic function and choose the most appropriate method for solving the equation. It also provides a deeper understanding of the graphical representation of the function and its relationship to the x-axis. So, remember this handy tool – it'll serve you well in your mathematical adventures!

Real-World Applications of Finding Zeros

Finding the zeros of a function isn't just a theoretical exercise; it has practical applications in many fields. Let's explore a few real-world scenarios where this skill comes in handy.

1. Physics: In physics, the zeros of a function can represent crucial points in a system's behavior. For example, when analyzing the trajectory of a projectile, the zeros of the height function indicate when the projectile hits the ground. Understanding these points allows physicists to predict the range and landing point of the projectile. Similarly, in circuit analysis, finding the zeros of a transfer function helps determine the resonant frequencies of a circuit, which are critical for designing filters and amplifiers.

2. Engineering: Engineers use zeros extensively in various applications. In structural engineering, the zeros of a function describing the deflection of a beam can identify points of maximum stress or instability. In control systems, the zeros of a system's transfer function influence the system's stability and response characteristics. Engineers manipulate these zeros to design systems that meet specific performance requirements. Moreover, in signal processing, zeros are used in filter design to attenuate or eliminate unwanted frequency components from a signal.

3. Economics: Economists use zeros to analyze market equilibrium and break-even points. For instance, the zeros of a profit function can represent the production levels at which a company breaks even, meaning its total revenue equals its total costs. Understanding these break-even points is crucial for making informed business decisions. In supply and demand models, the zeros of the excess demand function can indicate market equilibrium prices, where the quantity demanded equals the quantity supplied. These equilibrium points are vital for understanding market dynamics and predicting price fluctuations.

4. Computer Science: In computer graphics and game development, finding the zeros of functions is essential for collision detection. By determining when the distance between two objects is zero, developers can detect collisions and implement appropriate responses. In optimization algorithms, zeros can represent optimal solutions to problems, such as finding the minimum or maximum of a cost function. These optimization techniques are widely used in machine learning, data analysis, and other areas of computer science.

5. Finance: In finance, zeros are used to calculate rates of return and investment horizons. For example, the zeros of a present value function can determine the time it takes for an investment to reach a specific target value. In portfolio management, zeros can represent points of indifference between different investment strategies. Financial analysts use these zeros to make informed investment decisions and manage risk.

These are just a few examples of how finding the zeros of a function is used in the real world. As you continue your mathematical journey, you'll encounter many more applications of this fundamental concept. The ability to find zeros algebraically is a valuable skill that will serve you well in various fields.

Conclusion

Alright, guys, we've reached the end of our journey into finding the zeros of the function f(x) = 5x² - x + 10 algebraically. We've covered a lot of ground, from understanding what zeros are to applying the quadratic formula, dealing with complex numbers, and exploring the significance of the discriminant. We even touched on real-world applications, showing how this seemingly abstract concept has practical implications in various fields.

The key takeaway here is that finding zeros algebraically is a powerful tool. It allows us to solve equations exactly, understand the behavior of functions, and make predictions about real-world systems. The quadratic formula, in particular, is a versatile tool that can handle any quadratic equation, even those with complex solutions.

Remember, practice makes perfect. The more you work with these concepts, the more comfortable and confident you'll become. So, keep practicing, keep exploring, and keep asking questions. Math is a journey, and every step you take brings you closer to a deeper understanding of the world around you. And who knows, maybe you'll be the one to discover the next groundbreaking application of finding zeros!

So, until next time, keep those mathematical gears turning, and have fun exploring the world of numbers!

The zeros are: (1/10 + (√199 / 10)i, 1/10 - (√199 / 10)i)