Finding The Zeros Of A Quadratic: A Step-by-Step Guide

Hey everyone! Today, we're diving into a fundamental concept in algebra: finding the zeros of a quadratic function. Specifically, we'll be tackling the function h(x) = 2x² - 8x - 10. Don't worry, it might sound intimidating, but I promise, it's totally manageable! This guide will break down the process step-by-step, making it super easy to understand. So, grab your pencils and let's get started. Understanding zeros is crucial, guys, because they represent where the graph of the function crosses the x-axis. This knowledge is super useful for solving various real-world problems. We'll explore different methods like factoring, completing the square, and using the quadratic formula. These tools are super valuable not only in this context but also for your future math endeavors. Ready to become zero-finding pros? Let's get to it!

What are Zeros of a Quadratic Function?

So, what exactly do we mean by "zeros"? In the context of a quadratic function, which takes the general form of ax² + bx + c, the zeros are the values of x for which h(x) = 0. Essentially, we're looking for the x-values where the parabola (the U-shaped curve that represents a quadratic function) intersects the x-axis. These points are also known as the roots or solutions of the quadratic equation. Understanding this is key because these points provide vital insights into the behavior of the quadratic function. The x-intercepts, where the curve crosses the x-axis, directly influence the solution set of the equation. Finding these zeros is akin to finding the places where the function's output is zero. This has implications in physics, engineering, and various other fields where understanding the points of equilibrium or critical values is essential. By finding the zeros, we can fully understand the shape and behavior of the function. For example, if you were to graph the quadratic function, the zeros would be where the graph touches or crosses the horizontal axis. Knowing this helps you visualize and interpret the function's behavior. Understanding the concept of zeros and the ability to find them is, therefore, a fundamental skill in algebra.

Why are Zeros Important?

The significance of finding the zeros extends far beyond just solving a math problem. These values offer critical information about the function's behavior and have practical implications in several fields. They are essential in solving real-world problems, such as finding the points where a projectile hits the ground, determining break-even points in business, or calculating optimal values in engineering. The zeros give you insights into the function's trajectory. Furthermore, they are used to analyze the stability of systems in engineering and physics, where understanding the equilibrium points is essential. Zeros help you find the intercepts of a quadratic equation. Knowing these values can provide crucial data about the graph of the function. For instance, in physics, the zeros might represent the time when a ball thrown in the air hits the ground. In business, they might indicate the point at which a company starts making a profit. From a graphical perspective, the zeros are the x-intercepts. Therefore, knowing these points is vital to creating an accurate graph of the quadratic equation. Basically, finding the zeros is essential because it gives you key insights into where a function intersects the x-axis, and where the value of a function turns to zero.

Methods for Finding Zeros

There are several methods for finding the zeros of a quadratic function. Let's explore the three most common ones: factoring, completing the square, and using the quadratic formula.

Factoring Method

Factoring is often the quickest method when it works. The goal here is to rewrite the quadratic expression as a product of two binomials. Let's apply this to our function, h(x) = 2x² - 8x - 10. First, we can factor out a 2 from each term, resulting in 2(x² - 4x - 5). Now, we need to factor the quadratic expression inside the parentheses. We look for two numbers that multiply to -5 and add up to -4. Those numbers are -5 and 1. So, we can rewrite the expression as 2(x - 5)(x + 1). To find the zeros, we set each factor equal to zero and solve for x. For (x - 5) = 0, we get x = 5. For (x + 1) = 0, we get x = -1. Therefore, the zeros of the function are x = 5 and x = -1. However, guys, factoring doesn't always work perfectly. Some quadratics cannot be easily factored, which is when we turn to other methods.

Completing the Square

Completing the square is a more versatile method that works for all quadratic functions. It involves manipulating the equation to create a perfect square trinomial. Starting with h(x) = 2x² - 8x - 10, let's first factor out the leading coefficient (2) from the x² and x terms: 2(x² - 4x) - 10. Next, we need to complete the square inside the parentheses. Take half of the coefficient of the x term (-4), square it ((-2)² = 4), and add and subtract it inside the parentheses: 2(x² - 4x + 4 - 4) - 10. Now, rewrite the first three terms inside the parentheses as a perfect square: 2((x - 2)² - 4) - 10. Distribute the 2: 2(x - 2)² - 8 - 10, which simplifies to 2(x - 2)² - 18. To find the zeros, set the equation to zero: 2(x - 2)² - 18 = 0. Then, solve for x: 2(x - 2)² = 18, (x - 2)² = 9, x - 2 = ±3, so x = 2 ± 3. This gives us x = 5 and x = -1, the same zeros we found through factoring. While it takes a bit more effort, completing the square always works.

Quadratic Formula

The quadratic formula is the most universal method, as it always provides the solution, regardless of whether the quadratic can be factored easily or not. The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a, where a, b, and c are the coefficients from the standard form ax² + bx + c. For our function, h(x) = 2x² - 8x - 10, a = 2, b = -8, and c = -10. Substituting these values into the formula, we get x = (8 ± √((-8)² - 4 * 2 * -10)) / (2 * 2), which simplifies to x = (8 ± √(64 + 80)) / 4, and further to x = (8 ± √144) / 4. This means x = (8 ± 12) / 4. So, x = (8 + 12) / 4 = 5, and x = (8 - 12) / 4 = -1. Again, we get the zeros x = 5 and x = -1. The quadratic formula is a super powerful tool, so it's worth memorizing!

Step-by-Step Solution Using the Quadratic Formula

Since the quadratic formula is generally the most reliable method, let's go through it in more detail. Remember the formula: x = (-b ± √(b² - 4ac)) / 2a.

1. Identify Coefficients: From h(x) = 2x² - 8x - 10, we identify a = 2, b = -8, and c = -10. This step is critical; misidentifying even one coefficient can throw off the entire solution.
2. Substitute Values: Plug the coefficients into the formula: x = (-(-8) ± √((-8)² - 4 * 2 * -10)) / (2 * 2). Be careful with the negative signs, guys; they're easy to mess up!
3. Simplify the Expression Under the Square Root: Calculate the discriminant, the part under the square root: (-8)² - 4 * 2 * -10 = 64 + 80 = 144. So now, we have x = (8 ± √144) / 4.
4. Solve for x: Calculate the square root: √144 = 12. So, we now have x = (8 ± 12) / 4. Separate this into two solutions: x = (8 + 12) / 4 and x = (8 - 12) / 4. This gives us x = 20 / 4 = 5 and x = -4 / 4 = -1.
5. State the Zeros: The zeros of the function h(x) = 2x² - 8x - 10 are x = 5 and x = -1. Always double-check your work by substituting the zeros back into the original equation to ensure they make the equation equal zero. This method is foolproof and a great skill to have in your mathematical toolkit.

Conclusion

And there you have it, guys! We've successfully found the zeros of the quadratic function h(x) = 2x² - 8x - 10 using multiple methods. Remember, the zeros represent the x-values where the function equals zero, and these points are also the x-intercepts of the graph. Knowing how to find the zeros is super essential for understanding and working with quadratic functions. Practice these methods, and you'll become a pro in no time! So, go out there and conquer those quadratic equations. And always remember, practice makes perfect! Keep up the great work, and don't hesitate to revisit these steps anytime you need a refresher. Good luck, and keep exploring the amazing world of mathematics! Hope you found this useful!