No Solution: Find The Range Of K For Quadratic Equations

Hey Plastik Magazine readers! Today, let's dive into a super interesting problem that combines algebra and a bit of thinking outside the box. We're going to figure out when a system of equations has absolutely no real solutions. Sounds like fun? Let's get started!

The Problem at Hand

We've got two equations here:

1. y = k
2. y = x^2 - 12x + 41

Where k is a positive constant. The big question is: if this system has no real solutions, what must be true about all the possible values of k? Let's break this down step by step.

Understanding the Equations

First, let's get our heads around what these equations represent. The first equation, y = k, is just a horizontal line. Think of it as a flat line that runs across the graph at a height of k. Simple enough, right?

The second equation, y = x^2 - 12x + 41, is a quadratic equation, which means it forms a parabola when you graph it. Parabolas are those U-shaped curves that can open upwards or downwards. In this case, because the coefficient of x^2 is positive (it's 1), the parabola opens upwards. This is super important to visualize!

Visualizing the Problem

Now, imagine these two equations on the same graph. The horizontal line y = k is trying to intersect with the parabola y = x^2 - 12x + 41. If the system has a real solution, that means the line and the parabola intersect at one or more points. The x and y coordinates of those intersection points are the solutions to the system. But, if the system has no real solutions, that means the line and the parabola never touch each other. They just exist separately on the graph, never meeting.

Completing the Square

To really understand when these two graphs don't intersect, we need to rewrite the quadratic equation in vertex form. This will tell us the lowest point of the parabola (the vertex). We do this by completing the square. Take the quadratic equation y = x^2 - 12x + 41. To complete the square:

1. Take half of the coefficient of x (which is -12), and square it: (-12 / 2)^2 = (-6)^2 = 36
2. Add and subtract this value inside the equation: y = x^2 - 12x + 36 - 36 + 41
3. Rewrite the first three terms as a perfect square: y = (x - 6)^2 - 36 + 41
4. Simplify: y = (x - 6)^2 + 5

Now we have the equation in vertex form: y = (x - 6)^2 + 5. This tells us that the vertex of the parabola is at the point (6, 5). And since the parabola opens upwards, the lowest point on the parabola is at y = 5. This is critical to solving the problem.

Finding the Condition for No Real Solutions

Okay, we're at the home stretch! Remember, the line y = k has to not intersect the parabola for there to be no real solutions. Since the lowest point of the parabola is at y = 5, if the horizontal line y = k is above this point, there will be intersections and thus real solutions. But if the line y = k is below this point, there will be no intersections.

Since k is a positive constant, the line y = k must be below the vertex's y-value for no intersection. Therefore, k must be less than 5. So, the condition for no real solutions is k < 5.

Therefore, the correct answer is B: k < 5.

Deeper Dive: Why Completing the Square Matters

Completing the square is a technique used to rewrite a quadratic expression in a more useful form. For a quadratic equation in the form ax^2 + bx + c, completing the square allows us to rewrite it in the vertex form: a(x - h)^2 + k, where (h, k) is the vertex of the parabola. In our case, it helped us find the vertex (6, 5).

The vertex is a critical point on the parabola. It’s either the minimum (if the parabola opens upwards) or the maximum (if the parabola opens downwards) value of the quadratic function. Knowing the vertex makes it much easier to analyze the behavior of the quadratic and determine when it intersects with other functions, like the horizontal line y = k in our problem.

Without completing the square, you might try to use the discriminant (b^2 - 4ac) from the quadratic formula to determine when there are no real roots. However, that approach would require setting up a more complex inequality and might not be as intuitive for this particular problem.

By converting the quadratic to vertex form, we immediately identified the minimum y-value of the parabola. This made it straightforward to compare with the value of k and determine the condition for no intersections, and hence, no real solutions. This technique is invaluable for solving a wide variety of quadratic-related problems and provides a strong visual and intuitive understanding of what's happening graphically.

Why Other Options Are Wrong

Let's quickly go over why the other options are incorrect:

• A: k > 5 - If k is greater than 5, the horizontal line y = k will intersect the parabola at two points, meaning there are real solutions.
• C: k ≥ 5 - If k is equal to 5, the horizontal line y = k will intersect the parabola at exactly one point (the vertex), meaning there is one real solution. If k is greater than 5, there are two real solutions.
• D: 0 ≤ k ≤ 5 - While values of k between 0 and 5 (inclusive) might seem plausible, k = 5 results in one real solution. The problem specifies that there should be no real solutions.

Real-World Applications

While this might seem like an abstract math problem, these kinds of concepts actually pop up in various real-world scenarios. For example:

• Physics: Projectile motion can be modeled by quadratic equations. Determining when a projectile will not reach a certain height involves similar analysis.
• Engineering: Designing structures or systems where certain parameters must not exceed a critical threshold uses similar mathematical principles.
• Economics: Modeling cost curves or profit margins often involves quadratic equations. Finding conditions where costs do not exceed a certain level is a related problem.

Understanding how quadratic equations behave and when they have real solutions (or don't) is a valuable skill in many fields.

Conclusion

So, there you have it, Plastik Magazine crew! By completing the square and visualizing the graphs, we found that for the system of equations to have no real solutions, k must be less than 5 (k < 5). Always remember to break down complex problems into smaller, manageable steps, and don't be afraid to visualize what's going on. Keep practicing, and you'll become math whizzes in no time!

Until next time, keep those brains buzzing!