Finding Zeros: F(x) = 2x^2 - 10x - 3 Explained
Let's dive into the world of quadratic functions and explore how to find their zeros! In this article, we'll break down the process step-by-step, using the example function f(x) = 2x^2 - 10x - 3. Whether you're a student tackling algebra or just brushing up on your math skills, you'll find this guide helpful and easy to follow. Understanding zeros is crucial because these points reveal where the function's graph intersects the x-axis, providing key insights into the function's behavior and solutions. So, let's roll up our sleeves and get started on unraveling this mathematical puzzle.
Understanding Quadratic Functions
Before we jump into finding the zeros, let's make sure we're all on the same page about what a quadratic function actually is. A quadratic function is a polynomial function of the second degree, meaning the highest power of the variable (usually 'x') is 2. The general form of a quadratic function is f(x) = ax^2 + bx + c, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. If 'a' were zero, the function would become linear, not quadratic.
In our specific example, f(x) = 2x^2 - 10x - 3, we can identify the coefficients as follows: a = 2, b = -10, and c = -3. These coefficients play a crucial role in determining the shape and position of the parabola (the graph of a quadratic function) and, importantly, the zeros of the function. The 'a' coefficient determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0), while 'b' and 'c' influence the parabola's position on the coordinate plane. Understanding these basics is the first step towards mastering quadratic functions and their applications in various fields, from physics to engineering.
What are Zeros?
Now, let's talk about zeros. Zeros, also known as roots or x-intercepts, are the values of 'x' for which the function f(x) equals zero. In simpler terms, they are the points where the graph of the quadratic function crosses the x-axis. Finding these zeros is like uncovering the function's hidden secrets, as they provide valuable information about the function's behavior and solutions to related equations. Think of zeros as the function's 'sweet spots,' where the output is precisely zero.
Graphically, the zeros are where the parabola intersects the x-axis. A quadratic function can have two distinct real zeros, one repeated real zero, or no real zeros at all. This depends on how the parabola sits in relation to the x-axis. If the parabola crosses the x-axis at two distinct points, there are two real zeros. If the parabola touches the x-axis at only one point (the vertex), there is one repeated real zero. And if the parabola doesn't touch the x-axis at all, there are no real zeros (though there are complex zeros, which we won't delve into in this article). Understanding the concept of zeros is crucial not only in mathematics but also in practical applications, such as determining the trajectory of a projectile or optimizing a business process.
Methods to Find Zeros: The Quadratic Formula
There are a few ways to find the zeros of a quadratic function, but one of the most reliable and widely used methods is the quadratic formula. This formula is a powerful tool that provides a direct solution for the zeros, regardless of the complexity of the quadratic equation. It's like having a universal key that unlocks the zeros of any quadratic function. The quadratic formula is derived from the process of completing the square and is expressed as follows:
x = (-b ± √(b^2 - 4ac)) / 2a
Where 'a', 'b', and 'c' are the coefficients of the quadratic function f(x) = ax^2 + bx + c. This formula might look a bit intimidating at first, but once you break it down and practice using it, you'll find it's quite straightforward. The ± symbol indicates that there are typically two solutions: one where you add the square root term and one where you subtract it. The expression inside the square root, b^2 - 4ac, is called the discriminant, and it plays a crucial role in determining the nature of the zeros (whether they are real and distinct, real and repeated, or complex).
Applying the Quadratic Formula to f(x) = 2x^2 - 10x - 3
Now, let's put the quadratic formula into action and find the zeros of our function, f(x) = 2x^2 - 10x - 3. We've already identified the coefficients: a = 2, b = -10, and c = -3. The next step is to carefully substitute these values into the quadratic formula. It's crucial to pay close attention to the signs (positive and negative) to avoid errors.
Substituting the values, we get:
x = (-(-10) ± √((-10)^2 - 4 * 2 * -3)) / (2 * 2)
Now, let's simplify this step by step. First, we simplify the double negative in the first term and the multiplication in the denominator:
x = (10 ± √(100 + 24)) / 4
Next, we simplify the expression inside the square root:
x = (10 ± √124) / 4
Now we need to simplify the square root. We can factor 124 as 4 * 31, so √124 = √(4 * 31) = 2√31. Substituting this back into the equation, we get:
x = (10 ± 2√31) / 4
Finally, we can simplify this expression by dividing both terms in the numerator by 2:
x = (5 ± √31) / 2
The Zeros Found!
So, we've successfully found the zeros of the quadratic function f(x) = 2x^2 - 10x - 3! The zeros are:
x = (5 + √31) / 2 and x = (5 - √31) / 2
These are the two points where the parabola representing the function intersects the x-axis. We can also express these zeros in a slightly different form, separating the fraction:
x = 5/2 + √31/2 and x = 5/2 - √31/2
This form makes it clear that the zeros are centered around 5/2, with one zero shifted by √31/2 to the right and the other shifted by the same amount to the left. These exact values provide a precise understanding of where the function equals zero. Knowing the zeros allows us to not only visualize the parabola's position but also solve related quadratic equations and inequalities.
Comparing with the Options
Now, let's compare our solutions with the options provided in the original question. Our solutions are:
x = 5/2 + √31/2 and x = 5/2 - √31/2
The options given were:
A. x = -5/2 - √31/2 and x = -5/2 + √31/2 B. x = -5/2 - √(37/8) and x = -5/2 + √(37/8)
It's clear that our solutions do not match either of the provided options directly. Option A has the wrong sign for the 5/2 term, and Option B has a different value under the square root. This highlights the importance of careful calculation and verification when solving mathematical problems. If these were multiple-choice options in a test, this situation would suggest that there might be an error in the options provided or in the initial problem statement. Always double-check your work and compare your answers with the given options, but also be prepared to recognize when the options themselves might be incorrect.
Conclusion
In this article, we've walked through the process of finding the zeros of the quadratic function f(x) = 2x^2 - 10x - 3 using the quadratic formula. We started with an understanding of quadratic functions and the concept of zeros, then carefully applied the quadratic formula, simplified the expression, and arrived at the solutions: x = (5 + √31) / 2 and x = (5 - √31) / 2. We also compared our solutions with the provided options and noted that they didn't match, emphasizing the importance of accuracy and critical evaluation in problem-solving.
Understanding how to find zeros is a fundamental skill in algebra and calculus. It opens the door to solving a wide range of problems, from simple equations to complex real-world applications. Mastering the quadratic formula is a valuable asset in your mathematical toolkit, empowering you to tackle quadratic functions with confidence.
To further explore quadratic functions and their applications, you can visit resources like Khan Academy's Quadratic Equations Section. Keep practicing, and you'll become a pro at finding those zeros!
