Find The Zeros Of F(x) = X² - 18x - 4 A Step-by-Step Guide
Hey everyone! Today, we're going to dive into finding the zeros of the quadratic function f(x) = x² - 18x - 4. Finding the zeros, also known as roots or x-intercepts, means we're looking for the values of x that make the function equal to zero. In other words, we want to solve the equation x² - 18x - 4 = 0. This is a classic problem in algebra, and there are a couple of ways we can tackle it. We'll primarily focus on using the quadratic formula because it's a reliable method that works for any quadratic equation, even those that are difficult or impossible to factor. But before we jump into the quadratic formula, let's briefly touch on why factoring might not be the best approach in this specific case, and also, explore the method of completing the square as it provides a deeper understanding of the structure of quadratic equations and can be very useful in various mathematical contexts. Understanding these methods not only helps in solving for zeros but also enhances your overall problem-solving skills in mathematics. So, let’s get started and break down this problem step by step!
Why Factoring Might Not Be Straightforward
When faced with a quadratic equation, many of us instinctively try factoring first. Factoring is a great technique when it works because it's usually the quickest way to find the zeros. However, not all quadratic equations can be factored easily, especially if the roots are irrational or complex numbers. In this case, we're looking for two numbers that multiply to -4 and add up to -18. Think about the factors of -4: we have -1 and 4, 1 and -4, and -2 and 2. None of these pairs add up to -18, which suggests that the quadratic equation x² - 18x - 4 = 0 doesn't factor nicely using integers. This is a crucial observation because it guides us towards using a more general method, like the quadratic formula, which is designed to handle any quadratic equation, regardless of whether it's factorable or not. Recognizing when factoring is not the most efficient approach is a key skill in algebra. It saves time and prevents frustration. By understanding the limitations of factoring, we can appreciate the power and versatility of other methods, such as the quadratic formula and completing the square, which we will explore further in this discussion. So, while factoring is a valuable tool, it's important to have other techniques in your arsenal for those times when factoring just won't cut it.
Completing the Square: A Quick Detour
Before we dive headfirst into the quadratic formula, let’s take a quick detour to discuss another powerful technique: completing the square. While the quadratic formula offers a direct route to the zeros of a quadratic equation, completing the square provides a more insightful understanding of the equation's structure. It’s a method that transforms the quadratic expression into a perfect square trinomial, which can then be easily solved. Completing the square involves manipulating the equation to create a squared term, making it easier to isolate x. It’s particularly useful for deriving the quadratic formula itself and for converting quadratic equations into vertex form, which reveals the vertex of the parabola (the minimum or maximum point of the graph). Although it might seem a bit more involved than factoring or directly applying the quadratic formula, completing the square offers a deeper understanding of quadratic functions and their properties. This understanding can be invaluable in various mathematical contexts, such as calculus and analytic geometry. So, while we'll primarily use the quadratic formula to solve our specific problem, understanding completing the square adds another valuable tool to your mathematical toolkit. It’s like knowing the inner workings of an engine, not just how to drive the car. Let's briefly look at how this method would apply to our equation, giving us a broader perspective on solving quadratic equations.
Applying the Quadratic Formula
Okay, guys, since factoring isn't the most straightforward approach here, and we've had a glimpse of completing the square, let's use the quadratic formula. This formula is a powerful tool for finding the zeros of any quadratic equation in the form ax² + bx + c = 0. The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
In our case, we have f(x) = x² - 18x - 4, so we can identify the coefficients as a = 1, b = -18, and c = -4. Now, let's plug these values into the quadratic formula and see what we get. This is where the magic happens! We carefully substitute the values into the formula, making sure to pay close attention to the signs. The negative signs in the formula and in our coefficients can sometimes be tricky, so double-checking is always a good idea. Once we've substituted the values, we simplify the expression step by step, following the order of operations. This involves squaring numbers, multiplying, and then dealing with the square root. The expression under the square root, b² - 4ac, is particularly important because it's called the discriminant. The discriminant tells us about the nature of the roots – whether they are real or complex, and whether they are distinct or repeated. By understanding the role of the discriminant, we gain a deeper insight into the behavior of quadratic equations. So, let's go ahead and substitute, simplify, and uncover the zeros of our quadratic function!
Let's substitute a = 1, b = -18, and c = -4 into the formula:
x = (-(-18) ± √((-18)² - 4 * 1 * -4)) / (2 * 1)
Now, let's simplify this step by step. First, we deal with the negatives and the exponent:
x = (18 ± √(324 + 16)) / 2
Next, we add the numbers under the square root:
x = (18 ± √340) / 2
Now, we need to simplify the square root. We look for perfect square factors of 340. We can write 340 as 4 * 85, so:
x = (18 ± √(4 * 85)) / 2 x = (18 ± 2√85) / 2
Finally, we can divide both terms in the numerator by 2:
x = 9 ± √85
The Zeros of the Function
Alright, we've made it to the finish line! We've successfully applied the quadratic formula and simplified the expression to find the zeros of the function f(x) = x² - 18x - 4. The zeros are the values of x that make the function equal to zero, and we've found them to be x = 9 + √85 and x = 9 - √85. These are the two points where the parabola represented by the quadratic function intersects the x-axis. It's important to note that these roots are irrational numbers, which means they cannot be expressed as a simple fraction. This is why factoring didn't work out nicely for us at the beginning. The square root of 85 is an irrational number, and it contributes to the irrationality of the roots. The fact that we have two distinct real roots tells us that the discriminant (b² - 4ac) was positive, which we saw when we calculated it to be 340. If the discriminant had been zero, we would have had one real root (a repeated root), and if it had been negative, we would have had two complex roots. So, by finding these zeros, we've not only solved the equation but also gained valuable information about the nature of the quadratic function and its graph. Understanding the relationship between the roots, the discriminant, and the graph of the quadratic function is a fundamental concept in algebra, and it's something that will continue to be useful as you progress in your mathematical studies.
So, the zeros of the function are:
x = 9 + √85, 9 - √85
Expressing the Solution
Now that we've found the zeros, the last step is to express them in the requested format. The problem asks us to simplify the answer, including any radicals, and to use integers or fractions for any numbers in the expression. We've already done that! Our zeros are 9 + √85 and 9 - √85, which are simplified as much as possible. The problem also asks us to use a comma to separate answers as needed. So, we can write our final answer as:
9 + √85, 9 - √85
And that's it! We've successfully found the zeros of the quadratic function f(x) = x² - 18x - 4 and expressed them in the required format. This process demonstrates the power of the quadratic formula and how it can be used to solve any quadratic equation, regardless of its factorability. Remember, guys, practice makes perfect, so keep working on these types of problems to build your skills and confidence. Understanding quadratic equations and their solutions is a crucial foundation for more advanced math topics, so mastering this concept is definitely worth the effort. Great job, and keep up the excellent work!
Conclusion
In conclusion, we successfully found the zeros of the quadratic function f(x) = x² - 18x - 4 using the quadratic formula. We identified the coefficients a, b, and c, plugged them into the formula, simplified the expression, and arrived at the zeros: 9 + √85 and 9 - √85. We also discussed why factoring wasn't a straightforward approach in this case and briefly touched on the method of completing the square, which provides a deeper understanding of quadratic equations. Expressing the solution in the required format, we presented the zeros as 9 + √85, 9 - √85. This exercise highlights the importance of having a versatile toolkit of methods for solving quadratic equations, with the quadratic formula being a reliable and powerful technique for all cases. Understanding these concepts is crucial for further studies in mathematics and related fields. Keep practicing, and you'll become a pro at solving quadratic equations in no time!
