Finding The Roots Of Quadratic Equation Y=x^2-10x+125

In the realm of mathematics, quadratic equations hold a position of fundamental importance. These equations, characterized by their distinctive parabolic curves when graphed, find widespread applications across diverse fields, ranging from physics and engineering to economics and computer science. Among the most crucial aspects of understanding quadratic equations lies the determination of their roots, which represent the points where the parabola intersects the x-axis. This article delves into the intricacies of finding the roots of a given quadratic equation, specifically focusing on the equation y = x^2 - 10x + 125. We will explore the underlying concepts, demonstrate the application of the quadratic formula, and guide you through the step-by-step process of arriving at the correct solution. This article aims to provide a comprehensive and accessible resource for students, educators, and anyone seeking to deepen their understanding of quadratic equations.

Understanding Quadratic Equations

Before we embark on the journey of finding the roots of our specific equation, it is essential to lay a solid foundation by understanding the fundamental nature of quadratic equations. A quadratic equation is a polynomial equation of the second degree, generally expressed in the standard form:

ax^2 + bx + c = 0

where 'a', 'b', and 'c' are constants, and 'x' represents the variable. The coefficient 'a' cannot be zero, as this would reduce the equation to a linear one. The roots of a quadratic equation, also known as the solutions or zeros, are the values of 'x' that satisfy the equation. Graphically, these roots correspond to the points where the parabola, the graphical representation of the quadratic equation, intersects the x-axis.

Quadratic equations can have two distinct real roots, one real root (a repeated root), or two complex roots. The nature of the roots is determined by the discriminant, a crucial parameter calculated as:

Discriminant (Δ) = b^2 - 4ac

The discriminant provides valuable insights into the nature of the roots:

• If Δ > 0: The equation has two distinct real roots, indicating that the parabola intersects the x-axis at two different points.
• If Δ = 0: The equation has one real root (a repeated root), signifying that the parabola touches the x-axis at a single point (the vertex).
• If Δ < 0: The equation has two complex roots, implying that the parabola does not intersect the x-axis.

The Quadratic Formula: A Powerful Tool

The quadratic formula stands as a cornerstone in the realm of solving quadratic equations. It provides a universal method for determining the roots of any quadratic equation, regardless of the nature of its coefficients. The quadratic formula is expressed as:

x = (-b ± √(b^2 - 4ac)) / 2a

This formula elegantly encapsulates the relationship between the coefficients of the quadratic equation and its roots. The ± symbol signifies that there are potentially two solutions, one obtained by adding the square root term and the other by subtracting it. The expression under the square root, b^2 - 4ac, is none other than the discriminant, which, as we discussed earlier, plays a pivotal role in determining the nature of the roots.

Applying the Quadratic Formula to Our Equation: y = x^2 - 10x + 125

Now, let's put our knowledge into action and apply the quadratic formula to find the roots of the equation y = x^2 - 10x + 125. To begin, we need to identify the coefficients 'a', 'b', and 'c'. By comparing our equation to the standard form ax^2 + bx + c = 0, we can readily discern that:

• a = 1
• b = -10
• c = 125

With the coefficients in hand, we can now substitute them into the quadratic formula:

x = (-(-10) ± √((-10)^2 - 4 * 1 * 125)) / (2 * 1)

Simplifying the expression, we get:

x = (10 ± √(100 - 500)) / 2

x = (10 ± √(-400)) / 2

Dealing with Complex Roots

Notice that the discriminant, the value under the square root, is negative (-400). This indicates that the equation has complex roots. To handle the square root of a negative number, we introduce the imaginary unit 'i', defined as the square root of -1 (i = √-1). Therefore, we can rewrite √(-400) as:

√(-400) = √(400 * -1) = √400 * √-1 = 20i

Substituting this back into our equation, we get:

x = (10 ± 20i) / 2

Simplifying to Obtain the Roots

Finally, we simplify the expression by dividing both the real and imaginary parts by 2:

x = 5 ± 10i

Thus, the roots of the quadratic equation y = x^2 - 10x + 125 are:

x = 5 + 10i and x = 5 - 10i

These are complex roots, meaning they have both a real part (5) and an imaginary part (10i and -10i). This confirms our earlier observation that the parabola represented by this equation does not intersect the x-axis.

Conclusion: Mastering Quadratic Equations

In this comprehensive guide, we have successfully navigated the process of finding the roots of the quadratic equation y = x^2 - 10x + 125. We began by establishing a solid understanding of quadratic equations, their standard form, and the significance of the discriminant in determining the nature of the roots. We then introduced the powerful quadratic formula, a versatile tool for solving any quadratic equation. By applying the quadratic formula to our specific equation, we encountered complex roots, which led us to delve into the realm of imaginary numbers and the imaginary unit 'i'. Through careful simplification, we arrived at the solutions x = 5 + 10i and x = 5 - 10i. This journey underscores the importance of a strong grasp of fundamental mathematical concepts and the ability to apply them systematically to solve complex problems. By mastering quadratic equations, you unlock a powerful tool that will serve you well in various mathematical and scientific endeavors. Understanding the concepts, remembering the formulas, and practicing consistently are the keys to success in the world of quadratic equations. Remember, each equation is a puzzle waiting to be solved, and with the right tools and techniques, you can conquer them all.

Therefore, the correct answer is C. x = 5 ± 10i