Finding Parabolas With One Real Solution Intersecting The Line Y=x-5

Jul 10, 2025 by ADMIN 69 views

Exploring Parabolas and Linear Intersections Finding One Real Solution

In mathematics, particularly in algebra and analytic geometry, the intersection of a parabola and a line is a classic problem that reveals insightful relationships between quadratic and linear equations. Specifically, when we consider a parabola and a line intersecting at exactly one point, it signifies a unique condition where the line is tangent to the parabola. This tangency condition implies that the quadratic equation formed by equating the equations of the parabola and the line has exactly one real solution. In this article, we will delve into the specifics of determining which parabola, among a given set, will have one real solution when intersected by the line y = x - 5. The problem at hand involves finding the conditions under which a parabola, represented by a quadratic equation, and a line, represented by a linear equation, intersect at a single point. This scenario translates to finding when the quadratic equation, obtained by setting the parabola and line equations equal to each other, has exactly one real root. The number of real roots of a quadratic equation is determined by its discriminant, which is the part of the quadratic formula under the square root sign, denoted as b^2 - 4ac. When the discriminant is equal to zero, the quadratic equation has exactly one real solution, indicating that the line is tangent to the parabola. Our approach will involve setting the equation of the given line equal to the equation of the parabola and then examining the discriminant of the resulting quadratic equation. By setting the discriminant to zero, we can find the specific conditions that the parabola must satisfy to have exactly one point of intersection with the line. This article aims to provide a comprehensive understanding of how to solve such problems, offering a detailed explanation of the mathematical concepts and techniques involved.

Understanding the Intersection of a Parabola and a Line

The intersection of a parabola and a line can yield three possible scenarios: no intersection, one point of intersection, or two points of intersection. These scenarios are directly related to the solutions of the quadratic equation formed when the equations of the parabola and the line are equated. To fully grasp these scenarios, it's crucial to understand the discriminant, b^2 - 4ac, which plays a pivotal role in determining the nature of the roots of a quadratic equation. When the discriminant is positive (b^2 - 4ac > 0), the quadratic equation has two distinct real roots, indicating that the line intersects the parabola at two distinct points. This means the line crosses the parabola at two different locations, showing a clear passage through the curve. On the other hand, when the discriminant is negative (b^2 - 4ac < 0), the quadratic equation has no real roots, meaning the line and the parabola do not intersect at all. The line runs parallel or never comes close enough to the curve, resulting in no common points. The most interesting case occurs when the discriminant is equal to zero (b^2 - 4ac = 0). In this scenario, the quadratic equation has exactly one real root, indicating that the line is tangent to the parabola. This tangency means the line touches the parabola at exactly one point, grazing the curve without crossing it. This condition is the focus of our problem, as we seek to find the parabola that intersects the line y = x - 5 at a single point. To solve this, we will equate the line and parabola equations, rearrange the terms to form a quadratic equation in the standard form (ax^2 + bx + c = 0), and then set the discriminant to zero. This will provide us with the condition necessary for the line to be tangent to the parabola. Understanding these relationships between the discriminant and the intersection scenarios is essential for solving problems involving parabolas and lines in analytic geometry. It allows us to predict the number of intersection points without explicitly solving the quadratic equation, simply by analyzing the discriminant. This method is not only efficient but also provides a deeper insight into the geometric relationship between the parabola and the line.

Determining the Condition for One Real Solution

To find the parabola that has one real solution with the line y = x - 5, we will follow a methodical approach centered on the discriminant of a quadratic equation. This method involves several key steps, each crucial to arriving at the correct solution. First, we need to set the equation of the parabola equal to the equation of the line. This step combines the two equations into a single equation, representing the x-coordinates of the intersection points. The general form of a parabola is y = ax^2 + bx + c, and the given line is y = x - 5. Equating these gives us ax^2 + bx + c = x - 5. Next, we rearrange the equation to bring all terms to one side, setting the equation equal to zero. This step is essential for transforming the equation into the standard quadratic form, which is ax^2 + (b - 1)x + (c + 5) = 0. The coefficients of this quadratic equation are critical for calculating the discriminant. The discriminant, as mentioned earlier, is given by the formula b^2 - 4ac. In our rearranged equation, the coefficients are a, (b - 1), and (c + 5). Thus, the discriminant becomes (b - 1)^2 - 4a(c + 5). For the line to intersect the parabola at exactly one point, the discriminant must be equal to zero. This condition signifies that the quadratic equation has exactly one real root, meaning the line is tangent to the parabola. So, we set the discriminant to zero: (b - 1)^2 - 4a(c + 5) = 0. This equation gives us the condition that the coefficients a, b, and c must satisfy for the parabola to have one real solution with the line y = x - 5. By solving this equation, we can determine the specific parabola that meets this criterion. This condition is a powerful tool for analyzing the relationship between parabolas and lines, allowing us to predict their intersection behavior without needing to graph them or solve for the intersection points directly. Understanding and applying this method is crucial for solving a variety of problems in algebra and analytic geometry.

Applying the Condition to the Parabola y = x^2 + x - 4

Now, let's apply the condition we derived to the specific parabola y = x^2 + x - 4 and see if it intersects the line y = x - 5 at exactly one point. This involves substituting the coefficients of the given parabola into the discriminant equation and checking if the equation holds true. The given parabola equation is y = x^2 + x - 4, which can be rewritten in the standard quadratic form as ax^2 + bx + c, where a = 1, b = 1, and c = -4. We will now substitute these values into the discriminant condition (b - 1)^2 - 4a(c + 5) = 0 that we established earlier. Substituting the values, we get: (1 - 1)^2 - 4(1)(-4 + 5) = 0. Simplifying this expression step by step is crucial for avoiding errors. First, we evaluate the term inside the parentheses: (1 - 1)^2 = 0^2 = 0. Next, we evaluate the second term: 4(1)(-4 + 5) = 4(1)(1) = 4. Now, we combine these results: 0 - 4 = -4. So, the discriminant for this parabola and line is -4. Since the discriminant -4 is not equal to zero, the condition for one real solution is not met. This means that the line y = x - 5 does not intersect the parabola y = x^2 + x - 4 at exactly one point. Instead, it either intersects at two points or does not intersect at all. To determine whether the line intersects at two points or not at all, we would need to further analyze the discriminant. A negative discriminant indicates no intersection, while a positive discriminant would indicate two intersection points. In this case, since the discriminant is negative, the line y = x - 5 does not intersect the parabola y = x^2 + x - 4. This exercise demonstrates the power of using the discriminant to quickly determine the nature of the intersection between a parabola and a line. It avoids the need to solve the quadratic equation explicitly, saving time and effort. This method is a fundamental tool in analytic geometry and is widely used in various mathematical and engineering applications.

Conclusion

In conclusion, determining whether a parabola and a line intersect at one point involves analyzing the discriminant of the quadratic equation formed by equating the two functions. This article has provided a comprehensive explanation of the mathematical principles behind this concept, focusing on the significance of the discriminant (b^2 - 4ac) in determining the nature of the solutions. A discriminant of zero indicates exactly one real solution, signifying that the line is tangent to the parabola. We applied this principle to the parabola y = x^2 + x - 4 and the line y = x - 5, and through our calculations, we found that the discriminant was not zero, indicating that the line does not intersect the parabola at exactly one point. Instead, the negative discriminant revealed that the line and parabola do not intersect at all. This methodical approach, involving setting the equations equal, rearranging into standard quadratic form, and analyzing the discriminant, is a powerful tool for solving problems related to the intersection of parabolas and lines. Understanding this process not only helps in solving mathematical problems but also enhances the comprehension of the relationships between algebraic equations and their graphical representations. The ability to predict the number of intersection points without explicitly solving the equations is a valuable skill in mathematics and its applications. This article has aimed to provide a clear and detailed explanation, making the concepts accessible and applicable for various problem-solving scenarios. The use of the discriminant is a fundamental concept in algebra and analytic geometry, and mastering it opens the door to solving more complex problems involving conic sections and linear equations.