Finding Parabolas With One Real Solution: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a cool problem: figuring out which parabola will only touch the line y = x - 5 at just one point. Think of it like this: we're looking for a parabola that kisses the line, rather than crossing it.

To crack this, we'll need to use some algebra magic, specifically the discriminant. Don't worry, it's not as scary as it sounds! Let's break down the problem and then solve it step by step. This is super helpful because it allows you to visualize how parabolas and lines can interact and also gives you some neat problem-solving skills you can use in other math challenges. Knowing this can help you in standardized tests, understanding the geometry behind quadratic equations, and even in some real-world modeling scenarios. We'll examine the relationship between the equation of a parabola, the equation of a line, and how their intersection points are determined. Understanding this concept can unlock a deeper appreciation for how different algebraic elements come together in a visual and functional way. So, let’s get started. In this exploration, we'll begin by stating the problem clearly, and then, move into how to solve it. We will also describe the methods, the important concepts that are involved, and ultimately arrive at the correct answer through a detailed and thoughtful process.

Understanding the Problem

Okay, so the question wants us to pinpoint a parabola from a list that only has one point in common with the line y = x - 5. Remember, a parabola is a U-shaped curve, and a line is, well, a straight line. They can intersect in zero points (never touch), one point (touching or tangent), or two points (crossing). We need to find the parabola that has only one intersection point with the given line. The core idea here is understanding how the discriminant, part of the quadratic formula, helps us determine the number of real solutions (intersection points).

When a line and a parabola intersect, the points of intersection are the solutions to the system of equations formed by their equations. We're looking for a situation where the discriminant of the resulting quadratic equation is zero, indicating one real solution. The discriminant basically tells us how many times the parabola and line meet. If the discriminant is positive, they meet twice. If it’s zero, they touch at one point (a tangent relationship). And if it’s negative, they don't meet at all. The discriminant, often represented as 'b² - 4ac,' comes from the quadratic formula and is key to knowing how many solutions a quadratic equation has. The discriminant is at the heart of our solution, guiding us in the number of real roots. So, our aim is to find which parabola gives us a zero discriminant when we set up the equation with the line y = x - 5. This means the line is exactly touching the parabola at one point. The value of the discriminant will decide whether the solutions are real and distinct, repeated, or complex. This concept is fundamental when studying the properties of quadratic equations, and its ability to predict the nature of solutions without actually solving the quadratic equation is super useful.

The Method: Using the Discriminant

Here’s the game plan: We'll take each parabola, set its equation equal to the line's equation (y = x - 5), and rearrange everything to get a standard quadratic equation in the form of ax² + bx + c = 0. Then, we calculate the discriminant (b² - 4ac). If the discriminant equals zero, we've found our parabola!

This method hinges on understanding the quadratic formula and the information contained in the discriminant, a crucial part of the quadratic formula. The beauty of this approach lies in its systematic nature. It allows us to directly test the given parabolas against the line, avoiding complicated graphical analyses. By focusing on the algebraic properties, we're able to determine the exact number of intersection points without having to visualize the graphs. The ability to use the discriminant to predict the intersection is an incredibly powerful tool. It allows us to bypass drawing or graphing the equations, saving time and offering an algebraic shortcut. This ability to convert a geometric problem into a purely algebraic one underscores the strength of mathematical tools. This is a very useful approach for this kind of problem.

Step-by-Step Solution

Alright, let’s get down to brass tacks and apply this to our problem. We'll go through the given options one by one, and check the values.

1. Parabola 1: y = x² + x - 4

   • Set the equations equal: x² + x - 4 = x - 5
   • Rearrange into standard quadratic form: x² + x - x - 4 + 5 = 0 which simplifies to x² + 1 = 0
   • Identify a, b, and c: a = 1, b = 0, c = 1
   • Calculate the discriminant: b² - 4ac = 0² - 4 * 1 * 1 = -4

   Since the discriminant is negative (-4), this parabola does not have one real solution. The line and parabola do not intersect.

2. Parabola 2: y = u - v² + 2v - 1

   • First of all, let’s assume the equation is y = -x² + 2x - 1 for consistency since the equation's formatting has an error.
   • Set the equations equal: -x² + 2x - 1 = x - 5
   • Rearrange into standard quadratic form: -x² + 2x - x - 1 + 5 = 0 which simplifies to -x² + x + 4 = 0
   • Identify a, b, and c: a = -1, b = 1, c = 4
   • Calculate the discriminant: b² - 4ac = 1² - 4 * -1 * 4 = 1 + 16 = 17

   Since the discriminant is positive (17), this parabola does not have one real solution. The line and parabola intersect twice.

Conclusion

So, after working through the steps, it looks like there were some errors. Based on our calculations, neither of the parabolas given in the problem statement has only one real solution with the line y = x - 5. We aimed for a discriminant of zero to indicate one real solution (tangency), but neither option resulted in a zero discriminant. This highlights the importance of going through the step-by-step process of calculating and understanding the discriminant's role in determining the nature of the solutions.

This exploration underscores the importance of the discriminant in understanding quadratic equations and their relationships with lines. The method we followed is applicable to a broad range of related problems. Remember that the discriminant is your best friend when you are dealing with the number of solutions a quadratic equation will have. Keep practicing, and you'll get the hang of it in no time!

This is a super important concept in algebra and really helps you understand how different equations and graphs interact. Now that you have learned how to use the discriminant, you're well-equipped to tackle similar problems with confidence. Keep practicing and exploring, and happy calculating!