Solving Quadratic Equations Finding Roots Of Y = X^2 - 10x + 125
This article will walk you through the process of finding the roots of the quadratic equation $y = x^2 - 10x + 125$. We'll explore the quadratic formula, its application, and the significance of the results. We aim to clarify the steps involved and provide a comprehensive understanding of how to solve such problems. You'll not only learn how to find the roots but also gain a deeper appreciation for the underlying mathematical principles. Quadratic equations are fundamental in algebra and have numerous applications in various fields, making this a crucial skill to master. So, let's dive in and break down the solution step by step.
Understanding Quadratic Equations
Before we dive into solving the specific equation, let's briefly review what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is: $ax^2 + bx + c = 0$, where a, b, and c are constants, and a is not equal to zero. The roots of a quadratic equation are the values of x that satisfy the equation, meaning they are the values that make the equation true when substituted for x. These roots are also sometimes called solutions or zeros of the equation. Finding the roots is a common problem in algebra, and there are several methods to do so, including factoring, completing the square, and using the quadratic formula. Each method has its strengths and weaknesses, but the quadratic formula is generally the most versatile as it can be applied to any quadratic equation, regardless of whether it can be easily factored. In the given equation, $y = x^2 - 10x + 125$, we can identify that a = 1, b = -10, and c = 125. Now, with these values in mind, we can proceed to use the quadratic formula to find the roots. This formula provides a direct way to calculate the roots based on the coefficients of the quadratic equation. Understanding the nature of quadratic equations and their general form is crucial for solving them effectively.
The Quadratic Formula: The Key to Solving
The quadratic formula is a powerful tool for finding the roots of any quadratic equation. It states that for an equation of the form $ax^2 + bx + c = 0$, the roots x are given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula might look intimidating at first, but it's a straightforward way to solve quadratic equations once you understand its components. The term inside the square root, $b^2 - 4ac$, is called the discriminant. The discriminant is particularly important because it tells us about the nature of the roots. If the discriminant is positive, the equation has two distinct real roots. If it's zero, the equation has exactly one real root (a repeated root). And if the discriminant is negative, the equation has two complex roots. Complex roots involve the imaginary unit i, where $i = \sqrt{-1}$. In our equation, $y = x^2 - 10x + 125$, we've already identified that a = 1, b = -10, and c = 125. Now, we can plug these values into the quadratic formula to find the roots. This formula ensures that we can solve any quadratic equation, regardless of its complexity. Understanding and applying the quadratic formula is a cornerstone of algebra and is essential for solving various mathematical problems. By mastering this formula, you'll be well-equipped to tackle any quadratic equation you encounter.
Applying the Quadratic Formula to Our Equation
Now, let's apply the quadratic formula to the equation $y = x^2 - 10x + 125$. We have a = 1, b = -10, and c = 125. Plugging these values into the formula, we get: $x = \frac-(-10) \pm \sqrt{(-10)^2 - 4(1)(125)}}{2(1)}$. First, simplify the expression}2}$. Next, we calculate the discriminant}2}$. We can rewrite the square root of -400 as = \sqrt400 \cdot -1} = \sqrt{400} \cdot \sqrt{-1} = 20i$. So, the equation becomes{2}$. Finally, we simplify by dividing both terms in the numerator by 2: $x = 5 \pm 10i$. Thus, the roots of the quadratic equation are $x = 5 + 10i$ and $x = 5 - 10i$. This step-by-step application of the quadratic formula demonstrates how to handle complex roots and arrive at the correct solution. Each step is crucial to ensure accuracy, and understanding the process allows for efficient problem-solving.
Analyzing the Roots: Complex Numbers
The roots we found, $x = 5 \pm 10i$, are complex numbers. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, with $i^2 = -1$. The a part is called the real part, and the bi part is called the imaginary part. In our case, the roots are $5 + 10i$ and $5 - 10i$. Here, 5 is the real part, and 10i and -10i are the imaginary parts. These roots are a conjugate pair, meaning they have the same real part but opposite imaginary parts. Complex roots occur when the discriminant ($b^2 - 4ac$) of the quadratic equation is negative, as we saw in our calculation. The presence of complex roots indicates that the parabola represented by the quadratic equation does not intersect the x-axis. This is because the roots represent the x-intercepts of the graph. Complex numbers are fundamental in many areas of mathematics and physics, including electrical engineering, quantum mechanics, and signal processing. Understanding complex roots helps to provide a complete picture of the solutions to quadratic equations and their graphical representations. Recognizing and interpreting complex roots is an essential skill in algebra and beyond.
Conclusion: The Roots and Their Significance
In conclusion, by applying the quadratic formula to the equation $y = x^2 - 10x + 125$, we found the roots to be $x = 5 \pm 10i$. These roots are complex numbers, indicating that the parabola represented by the equation does not intersect the x-axis. The quadratic formula is a versatile tool for solving quadratic equations, regardless of whether the roots are real or complex. Understanding the discriminant helps determine the nature of the roots, which is a crucial step in solving these equations. Quadratic equations and their roots have significant applications in various fields, including physics, engineering, and computer science. Being able to solve them accurately is a fundamental skill in mathematics. The process we've outlined provides a clear and systematic approach to finding the roots of quadratic equations, ensuring a solid understanding of the underlying principles and their implications. Mastering these concepts opens doors to more advanced mathematical topics and real-world applications, making it a valuable asset in your mathematical toolkit. We hope this detailed explanation has clarified the process and provided you with a comprehensive understanding of finding the roots of quadratic equations.
