Nature Of Solutions For Quadratic Equation Y=x²-11x+7
Understanding the nature of solutions for quadratic equations is a fundamental concept in algebra. Quadratic equations, characterized by the general form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0, play a crucial role in various mathematical and scientific applications. The solutions, also known as roots or zeros, represent the values of x that satisfy the equation, making the expression equal to zero. In this article, we delve into the quadratic equation y = x² - 11x + 7, meticulously examining its characteristics to determine the true nature of its solutions. We'll explore the significance of the discriminant, a pivotal component in classifying the solutions, and employ it to unveil whether the equation yields real or complex solutions, and how many of each type exist. By the end of this exploration, you will have a solid understanding of how to analyze quadratic equations and interpret the solutions they produce.
Decoding the Discriminant: The Key to Unlocking Solution Types
The discriminant serves as a compass, guiding us through the landscape of quadratic equation solutions. This powerful tool, denoted as Δ (Delta), is calculated using the formula Δ = b² - 4ac, derived from the coefficients of the quadratic equation ax² + bx + c = 0. The discriminant's value acts as a signpost, revealing the nature and number of solutions the equation possesses. When Δ is positive (Δ > 0), the equation boasts two distinct real solutions, signifying that the parabola intersects the x-axis at two unique points. This scenario suggests a diverse range of values for x that satisfy the equation, making it a common occurrence in many real-world applications. Conversely, a negative discriminant (Δ < 0) indicates the presence of two complex solutions, where the parabola never crosses the x-axis. These solutions involve the imaginary unit i (√-1), expanding the realm of solutions beyond the real number line. A discriminant of zero (Δ = 0) signals a single real solution, a special case where the parabola touches the x-axis at exactly one point, representing a repeated root. This understanding of the discriminant empowers us to predict the solution landscape of a quadratic equation without explicitly solving it, saving time and providing valuable insights.
Applying the Discriminant to Our Equation: y = x² - 11x + 7
Let's put the discriminant to work on our specific equation, y = x² - 11x + 7. By carefully comparing this equation to the general form ax² + bx + c = 0, we can identify the coefficients: a = 1, b = -11, and c = 7. Now, we'll substitute these values into the discriminant formula: Δ = b² - 4ac. This yields Δ = (-11)² - 4(1)(7) = 121 - 28 = 93. The result, 93, is a positive number. This crucial finding immediately tells us that the quadratic equation y = x² - 11x + 7 has two distinct real solutions. The positive discriminant confirms that the parabola represented by this equation intersects the x-axis at two different points. These points correspond to the two real values of x that satisfy the equation, showcasing the practical application of the discriminant in determining the nature of solutions.
Delving Deeper: Understanding Real Solutions
Real solutions, the familiar inhabitants of the number line, are the solutions we often encounter in everyday applications of mathematics. They can be integers, rational numbers, or irrational numbers, and they represent tangible values that can be plotted on a graph. When a quadratic equation has two real solutions, it means the graph of the equation, a parabola, crosses the x-axis at two distinct points. These points of intersection represent the values of x for which y equals zero. In the context of the equation y = x² - 11x + 7, the two real solutions signify that there are two specific x-values that make the equation true. To find these solutions, we can employ methods like factoring, completing the square, or the quadratic formula. The quadratic formula, x = (-b ± √(b² - 4ac)) / 2a, is particularly useful as it provides a direct way to calculate the solutions, utilizing the discriminant we've already computed. The real solutions of a quadratic equation have concrete interpretations in various scenarios, such as finding the points where a projectile hits the ground or determining the dimensions of a rectangular area given its perimeter and area.
Unveiling the Solutions with the Quadratic Formula
To explicitly find the two real solutions of y = x² - 11x + 7, we'll harness the power of the quadratic formula. As we've established, the coefficients are a = 1, b = -11, and c = 7, and the discriminant Δ is 93. Substituting these values into the quadratic formula, x = (-b ± √(b² - 4ac)) / 2a, we get:
x = (-(-11) ± √93) / 2(1)
x = (11 ± √93) / 2
This gives us two distinct solutions:
x₁ = (11 + √93) / 2
x₂ = (11 - √93) / 2
These two values represent the x-coordinates where the parabola intersects the x-axis, confirming our earlier deduction based on the positive discriminant. While these solutions are expressed in radical form, they are indeed real numbers, lying on the number line. Approximating these values using a calculator gives us a clearer sense of their magnitude and placement on the number line. This process demonstrates the practical application of the quadratic formula in finding the explicit solutions of a quadratic equation, building upon our understanding of the discriminant.
Discarding Complex Solutions: A Matter of the Discriminant
Complex solutions, unlike their real counterparts, venture beyond the familiar number line and into the realm of imaginary numbers. These solutions arise when the discriminant (Δ) of a quadratic equation is negative. A negative discriminant implies that the square root of a negative number is involved, leading to solutions that include the imaginary unit i (√-1). In the context of a parabola, complex solutions mean that the parabola does not intersect the x-axis. The roots are not real numbers; they are complex conjugates. Our analysis of y = x² - 11x + 7 revealed a positive discriminant (Δ = 93), definitively ruling out the possibility of complex solutions. This highlights the crucial role of the discriminant in determining the nature of solutions, allowing us to quickly identify whether a quadratic equation has real or complex roots without resorting to more elaborate methods. Understanding when complex solutions occur is essential in various mathematical contexts, particularly in fields like electrical engineering and quantum mechanics, where complex numbers play a significant role.
Why Complex Solutions Don't Apply to Our Equation
Since the discriminant of y = x² - 11x + 7 is positive (Δ = 93), the equation does not possess any complex solutions. Complex solutions emerge only when the discriminant is negative, indicating the presence of the imaginary unit i in the solutions. The absence of complex solutions in this case reinforces the idea that the parabola intersects the x-axis at two distinct points, corresponding to the two real solutions we calculated using the quadratic formula. This distinction between real and complex solutions is fundamental in understanding the behavior of quadratic equations and their graphical representations. By focusing on the discriminant, we can efficiently classify the nature of the solutions and avoid unnecessary calculations, streamlining the problem-solving process. The clarity provided by the discriminant allows us to confidently conclude that the solutions are real, providing a solid foundation for further analysis and application of the results.
Conclusion: The Two Real Solutions of y = x² - 11x + 7
In our comprehensive analysis of the quadratic equation y = x² - 11x + 7, we've successfully navigated the terrain of solution types and arrived at a definitive conclusion. By employing the discriminant (Δ = b² - 4ac), we determined that Δ = 93, a positive value. This positive discriminant unequivocally indicates that the equation has two distinct real solutions. We further validated this finding by applying the quadratic formula, which yielded the two real solutions: x₁ = (11 + √93) / 2 and x₂ = (11 - √93) / 2. These solutions represent the x-intercepts of the parabola, the points where the graph intersects the x-axis. The positive discriminant also allowed us to confidently discard the possibility of complex solutions, streamlining our analysis and focusing our efforts on the real number domain. This journey through the equation y = x² - 11x + 7 underscores the power of the discriminant as a tool for understanding the nature of quadratic equation solutions, enabling us to efficiently classify and calculate them. The ability to discern between real and complex solutions is crucial in various mathematical and scientific applications, making the discriminant an indispensable concept in the study of quadratic equations.
