Finding Integer Solutions Sum For 5x^2-2kx+1<0

Hey everyone! Today, we're diving into a super interesting problem that combines algebra, precalculus, number theory, and quadratics. We're going to figure out how to find the sum of all positive integers k that make the inequality 5x² - 2kx + 1 < 0 have exactly one integer solution. Sounds like a mouthful, right? But trust me, we'll break it down step by step so it's super easy to understand. Let's get started!

Understanding the Problem: Setting the Stage

So, our main goal here is to find those special positive integer values for k. These values, when plugged into the inequality 5x² - 2kx + 1 < 0, give us a quadratic expression that has only one integer solution. That means there's only one whole number that, when we substitute it for x, makes the inequality true. This is a classic problem that blends algebraic manipulation with some clever thinking about number properties, and it's the kind of question that really makes you think. It's not just about crunching numbers; it's about understanding the behavior of quadratic functions and how integers fit into the picture. We need to consider the quadratic formula, the discriminant, and how these relate to the roots of the quadratic. The roots are the values of x that make the quadratic expression equal to zero, and they play a crucial role in determining the interval where the inequality 5x² - 2kx + 1 < 0 holds true. If the interval is too narrow, it might not contain any integers. If it's too wide, it might contain more than one. We're looking for that sweet spot where it contains exactly one. We'll also be using our knowledge of the discriminant (that part under the square root in the quadratic formula) to understand how many real roots the quadratic has. If the discriminant is negative, there are no real roots, and our inequality won't have any solutions. If it's zero, there's exactly one real root, and if it's positive, there are two distinct real roots. All of this background knowledge is going to come together to help us solve this problem. So, buckle up, and let's start unpacking this!

Cracking the Quadratic: Finding the Roots

Okay, first things first, let's tackle that quadratic inequality: 5x² - 2kx + 1 < 0. To figure out when this is true, we need to find the roots of the corresponding quadratic equation, 5x² - 2kx + 1 = 0. Remember the quadratic formula? It's our trusty tool for solving these types of equations. The quadratic formula is essentially a recipe for finding the roots of any quadratic equation in the form ax² + bx + c = 0. It tells us that the roots, x, are given by x = (-b ± √(b² - 4ac)) / (2a). In our case, a = 5, b = -2k, and c = 1. So, let's plug these values into the quadratic formula and see what we get. x = (2k ± √((-2k)² - 4 * 5 * 1)) / (2 * 5) simplifies to x = (2k ± √(4k² - 20)) / 10. We can even simplify this a bit further by factoring out a 4 from under the square root: x = (2k ± 2√( k² - 5)) / 10. And finally, we can divide both the numerator and the denominator by 2 to get x = (k ± √(k² - 5)) / 5. These two values of x are the roots of our quadratic equation. They represent the points where the parabola defined by 5x² - 2kx + 1 intersects the x-axis. Now, since we're dealing with an inequality (5x² - 2kx + 1 < 0), we're interested in the interval between these roots. This is because the parabola opens upwards (since the coefficient of x² is positive), so the quadratic expression will be negative between the roots. Our next step is to understand how the values of k affect these roots and the interval between them. Remember, we're looking for the values of k that give us exactly one integer solution within this interval. This is where the real fun begins!

Analyzing the Roots: Intervals and Integers

Now that we've got the roots, x = (k ± √(k² - 5)) / 5, let's dig deeper into what they tell us. These roots define an interval on the x-axis, and it's within this interval that our inequality 5x² - 2kx + 1 < 0 holds true. Think of it like this: the roots are the boundaries, and the solution to the inequality is the space in between. The smaller root is (k - √(k² - 5)) / 5, and the larger root is (k + √(k² - 5)) / 5. Our inequality is satisfied for all x values between these two roots. The key to this problem is figuring out when there's exactly one integer within this interval. To do that, we need to understand how the length of the interval changes as k changes. The length of the interval is simply the difference between the larger and smaller roots: [(k + √(k² - 5)) / 5] - [(k - √(k² - 5)) / 5] = (2√(k² - 5)) / 5. This formula tells us that the length of the interval depends on the value of k. As k increases, the length of the interval also increases. Now, here's a crucial observation: for our inequality to have any real solutions, the expression under the square root, k² - 5, must be non-negative. In other words, k² - 5 ≥ 0. This means that k² ≥ 5, which implies that k must be greater than or equal to √5 (approximately 2.24). Since we're looking for positive integers, the smallest possible value for k is 3. But just because k is greater than or equal to 3 doesn't guarantee that we'll have exactly one integer solution. We need to carefully analyze how the roots and the interval change as we increase k. For small values of k, the interval might be very narrow, containing at most one integer. As k gets larger, the interval widens, potentially containing more than one integer. Our goal is to find the values of k where the interval is just wide enough to contain one integer and no more.

Finding the Right Fit: Exactly One Integer Solution

Now comes the tricky part: pinpointing the values of k that give us exactly one integer solution. We know the interval is defined by the roots (k ± √(k² - 5)) / 5, and we want only one integer to fall within this range. Let's think about how we can make this happen. Imagine the number line. Our interval is a segment on this line, and we want exactly one integer to be trapped inside. This means that the length of the interval must be less than 2. If it were 2 or greater, we'd have room for at least two integers. So, we have the condition (2√(k² - 5)) / 5 < 2. Let's simplify this inequality to get a better handle on the possible values of k. Multiplying both sides by 5, we get 2√(k² - 5) < 10. Dividing by 2, we have √(k² - 5) < 5. Now, we can square both sides to get rid of the square root: k² - 5 < 25. Adding 5 to both sides gives us k² < 30. This tells us that k must be less than √30 (approximately 5.48). Remember, we also know that k must be an integer greater than or equal to 3 (from our earlier analysis of k² - 5 ≥ 0). So, the possible values for k are 3, 4, and 5. But we're not done yet! We need to check each of these values to make sure they actually give us exactly one integer solution. This is crucial because the inequality k² < 30 only gives us an upper bound for k. It doesn't guarantee that every k in the range 3, 4, and 5 will work. We need to plug each value back into our original inequality and see what happens.

The Final Check: Verifying the Solutions

Okay, we've narrowed it down to three potential values for k: 3, 4, and 5. Now, let's put these to the test and see if they actually work. This is where the rubber meets the road, guys! We need to plug each value of k back into the inequality 5x² - 2kx + 1 < 0 and see if it has exactly one integer solution. Let's start with k = 3. Our inequality becomes 5x² - 6x + 1 < 0. We can factor the quadratic expression as (5x - 1)(x - 1) < 0. This inequality holds true when 1/5 < x < 1. The only integer in this interval is none, so k = 3 doesn't work. Next, let's try k = 4. The inequality becomes 5x² - 8x + 1 < 0. Using the quadratic formula, the roots are x = (4 ± √(16 - 5)) / 5 = (4 ± √11) / 5. Approximately, these roots are (4 - 3.32) / 5 ≈ 0.14 and (4 + 3.32) / 5 ≈ 1.46. The integers in this interval are 1, so k = 4 works! Finally, let's check k = 5. The inequality becomes 5x² - 10x + 1 < 0. The roots are x = (5 ± √(25 - 5)) / 5 = (5 ± √20) / 5 = (5 ± 2√5) / 5. Approximately, these roots are (5 - 4.47) / 5 ≈ 0.11 and (5 + 4.47) / 5 ≈ 1.89. The integers in this interval are 1, so k = 5 works too! So, we've found two values of k that satisfy our condition: k = 4 and k = 5. The final step is to find the sum of these values. 4 + 5 = 9. And there you have it! The sum of all positive integers k for which 5x² - 2kx + 1 < 0 has exactly one integral solution is 9. This problem is a great example of how different areas of math can come together to solve a single question. We used the quadratic formula, analyzed inequalities, and thought carefully about integers and intervals. It's all about breaking down the problem into smaller, manageable steps and then putting the pieces back together. Great job, guys!

Final Answer: Summing It Up

So, after all that awesome math-ing, we've reached our final destination. We've identified the positive integers k that make the inequality 5x² - 2kx + 1 < 0 have exactly one integer solution. Remember, we started by using the quadratic formula to find the roots of the equation, which gave us the interval where the inequality holds true. Then, we analyzed how the length of this interval changes with different values of k. We figured out that the interval needs to be narrow enough to contain only one integer. We used some inequalities to narrow down the possible values of k to 3, 4, and 5. Finally, we put each of these values to the test, plugging them back into the original inequality and checking how many integer solutions we got. We found that k = 4 and k = 5 are the only values that work. The final step was super simple: we just added these values together to get our answer. Therefore, the sum of all positive integers k for which 5x² - 2kx + 1 < 0 has exactly one integral solution is 4 + 5 = 9. Isn't it amazing how a seemingly complex problem can be solved with a combination of different mathematical tools and a bit of logical thinking? This kind of problem really showcases the beauty and interconnectedness of mathematics. Keep practicing, keep exploring, and you'll be amazed at what you can achieve! Keep up the great work, everyone!