Solving The Quadratic Expression (-5 + √(5² - 4(3)(-2))) / (2(3))

Jul 10, 2025 by Admin 66 views

$Unraveling the Quadratic Solution $\frac{-5+\sqrt{5^2-4(3)(-2)}}{2(3)}$: A Comprehensive Guide$

Introduction

In the realm of mathematics, quadratic equations hold a significant place, serving as fundamental tools for modeling a myriad of real-world phenomena. These equations, characterized by their second-degree polynomial form, often require specific methods for finding their solutions, commonly known as roots or zeros. One of the most widely used techniques for solving quadratic equations is the quadratic formula. This formula provides a direct method for calculating the roots of any quadratic equation expressed in the standard form $ax^2 + bx + c = 0$ , where a, b, and c are coefficients. The expression $\frac{-5+\sqrt{5^2-4(3)(-2)}}{2(3)}$ is a direct application of the quadratic formula, designed to find one of the roots of a specific quadratic equation. In this comprehensive guide, we will meticulously dissect this expression, revealing its underlying mathematical principles and demonstrating its application in solving quadratic equations. We'll begin by contextualizing the quadratic formula itself, tracing its origins and highlighting its significance in mathematical problem-solving. Following this, we will delve into a step-by-step evaluation of the given expression, breaking down each component to ensure clarity and understanding. Our exploration will encompass simplifying the terms under the square root, performing the necessary arithmetic operations, and arriving at the final numerical solution. Furthermore, we will discuss the broader implications of this solution within the context of quadratic equations, interpreting its meaning in terms of the roots of the equation. By the end of this discussion, you will gain a thorough understanding of how to evaluate such expressions and appreciate their role in solving quadratic equations.

Understanding the Quadratic Formula

The quadratic formula is a cornerstone in algebra, offering a universal method for determining the roots of any quadratic equation. It is mathematically expressed as: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where a, b, and c are the coefficients of the quadratic equation in its standard form, $ax^2 + bx + c = 0$ . The quadratic formula is not just a random mathematical tool; it is derived from the method of completing the square, a technique used to transform a quadratic equation into a perfect square trinomial. This derivation underscores the logical foundation of the formula and highlights its inherent connection to the fundamental principles of algebra. The term inside the square root, $b^2 - 4ac$ , is known as the discriminant, and it plays a crucial role in determining the nature of the roots. If the discriminant is positive, the quadratic equation has two distinct real roots. If it is zero, the equation has exactly one real root (a repeated root). If it is negative, the equation has two complex roots. The $\pm$ symbol in the formula indicates that there are generally two solutions, one obtained by adding the square root term and the other by subtracting it. In our specific expression, $\frac{-5+\sqrt{5^2-4(3)(-2)}}{2(3)}$ , we can see the quadratic formula in action. By comparing this expression to the general form of the quadratic formula, we can identify the values of a, b, and c for the corresponding quadratic equation. This identification is the first step in understanding the context of the expression and its relationship to the roots of the equation. The quadratic formula's significance extends beyond pure mathematics; it is widely used in physics, engineering, economics, and computer science to model and solve problems involving parabolic trajectories, optimization, and other quadratic relationships. Its robustness and versatility make it an indispensable tool for anyone working with mathematical models.

Step-by-Step Evaluation of the Expression

To effectively evaluate the expression $\frac{-5+\sqrt{5^2-4(3)(-2)}}{2(3)}$ , we will proceed through a detailed, step-by-step breakdown. This approach ensures clarity and minimizes the potential for errors.

Simplify the term under the square root: The expression under the square root is $5^2 - 4(3)(-2)$ . We begin by evaluating the exponent: $5^2 = 25$ . Next, we multiply the terms: $4(3)(-2) = -24$ . Now we substitute these values back into the expression: $25 - (-24)$ . Subtracting a negative number is equivalent to adding its positive counterpart, so we have $25 + 24 = 49$ . Therefore, the expression under the square root simplifies to 49.
Evaluate the square root: Now that we have simplified the term under the square root, we can evaluate $\sqrt{49}$ . The square root of 49 is 7, since $7 * 7 = 49$ .
Substitute the square root value back into the expression: Replacing $\sqrt{49}$ with 7, our expression becomes $\frac{-5 + 7}{2(3)}$ .
Simplify the numerator: The numerator is $-5 + 7$ , which simplifies to 2.
Simplify the denominator: The denominator is $2(3)$ , which simplifies to 6.
Divide the numerator by the denominator: Now we have $\frac{2}{6}$ . This fraction can be further simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2. Dividing both by 2, we get $\frac{1}{3}$ .

Therefore, the value of the expression $\frac{-5+\sqrt{5^2-4(3)(-2)}}{2(3)}$ is $\frac{1}{3}$ . This step-by-step evaluation clearly demonstrates how to approach such expressions, breaking them down into manageable components and performing the necessary arithmetic operations. Understanding each step is crucial for mastering the evaluation of more complex mathematical expressions. The result, $\frac{1}{3}$ , represents one of the roots of the quadratic equation from which this expression was derived.

Contextualizing the Solution within Quadratic Equations

Now that we have meticulously evaluated the expression $\frac{-5+\sqrt{5^2-4(3)(-2)}}{2(3)}$ and arrived at the solution $\frac{1}{3}$ , it is essential to place this result within the context of quadratic equations. This involves identifying the specific quadratic equation from which this expression originated and understanding the significance of the solution in relation to the equation's roots. The given expression is a direct application of the quadratic formula, which, as we discussed earlier, is used to find the roots of a quadratic equation in the form $ax^2 + bx + c = 0$ . By comparing the expression to the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, we can deduce the values of the coefficients a, b, and c for the corresponding quadratic equation. In our expression, we have $\frac{-5+\sqrt{5^2-4(3)(-2)}}{2(3)}$ . Comparing this to the formula, we can identify: * -b = -5, which implies b = 5. * a = 3. * 4ac = 4(3)(-2), which implies c = -2.

Therefore, the quadratic equation from which this expression is derived is $3x^2 + 5x - 2 = 0$ . The solution $\frac{1}{3}$ represents one of the roots of this equation. Quadratic equations, being second-degree polynomials, generally have two roots. These roots are the values of x that satisfy the equation, meaning that when these values are substituted into the equation, the result is zero. To find the other root, we would use the other part of the quadratic formula, which involves subtracting the square root term: $x = \frac-b - \sqrt{b^2 - 4ac}}{2a}$ Substituting the values a = 3, b = 5, and c = -2 into this formula, we get $x = \frac{-5 - \sqrt{5^2 - 4(3)(-2)}{2(3)} = \frac{-5 - \sqrt{49}}{6} = \frac{-5 - 7}{6} = \frac{-12}{6} = -2$ So, the other root of the quadratic equation $3x^2 + 5x - 2 = 0$ is -2. The roots of a quadratic equation have a graphical interpretation as well. They represent the x-intercepts of the parabola described by the quadratic equation when plotted on a coordinate plane. In this case, the parabola represented by $3x^2 + 5x - 2 = 0$ intersects the x-axis at $x = \frac{1}{3}$ and x = -2. Understanding the contextual meaning of solutions is crucial in mathematical problem-solving. It allows us to connect abstract algebraic concepts to real-world applications and interpret the results in a meaningful way.

Real-World Applications and Significance

The ability to solve quadratic equations, as demonstrated through the evaluation of the expression $\frac{-5+\sqrt{5^2-4(3)(-2)}}{2(3)}$ , is not merely an academic exercise; it has profound implications and applications in various real-world scenarios. Quadratic equations are fundamental tools for modeling a wide array of phenomena across diverse fields, including physics, engineering, economics, and computer science. In physics, quadratic equations are used to describe projectile motion. The trajectory of an object thrown into the air, such as a ball or a rocket, can be accurately modeled using a quadratic equation. The roots of the equation represent the points where the object hits the ground, and the vertex of the parabola represents the maximum height reached by the object. Engineers use quadratic equations in structural design, particularly in calculating the forces and stresses on beams and arches. The shape of a parabolic arch, for example, is naturally described by a quadratic equation, and understanding the roots and vertex of this equation is crucial for ensuring the structural integrity of the arch. In economics, quadratic functions are used to model cost, revenue, and profit curves. The break-even points of a business, where revenue equals cost, can be found by solving a quadratic equation. The maximum profit can also be determined by finding the vertex of the profit curve. Computer scientists and game developers use quadratic equations in computer graphics and game physics. For instance, the path of a projectile in a video game can be simulated using a quadratic equation, making the game more realistic and engaging. The solution we obtained, $\frac{1}{3}$ , represents a specific instance of how quadratic equations can provide valuable insights into real-world problems. While this particular solution may not directly correspond to a specific physical quantity in a given scenario, the process of solving quadratic equations and interpreting their roots is universally applicable. The significance of understanding quadratic equations extends beyond these specific examples. It fosters a deeper understanding of mathematical modeling and problem-solving, equipping individuals with the tools to analyze and address complex challenges in a variety of fields. The quadratic formula, and the ability to evaluate expressions derived from it, is a cornerstone of mathematical literacy and a valuable asset in numerous professional domains.

Conclusion

In this comprehensive guide, we have thoroughly explored the expression $\frac{-5+\sqrt{5^2-4(3)(-2)}}{2(3)}$ , dissecting its components and revealing its connection to the fundamental principles of quadratic equations. We began by introducing the quadratic formula, a cornerstone of algebra, and highlighting its significance in solving equations of the form $ax^2 + bx + c = 0$ . We then meticulously evaluated the given expression, breaking it down into manageable steps, including simplifying the term under the square root, evaluating the square root, and performing the necessary arithmetic operations. This step-by-step approach not only allowed us to arrive at the solution, $\frac{1}{3}$ , but also provided a clear understanding of the underlying mathematical processes. Furthermore, we contextualized this solution within the broader context of quadratic equations, identifying the specific equation from which it originated and interpreting its meaning as one of the roots of the equation. We also discussed the graphical representation of the roots as x-intercepts of a parabola, reinforcing the visual and conceptual understanding of quadratic solutions. Finally, we explored the real-world applications of quadratic equations, showcasing their relevance in fields such as physics, engineering, economics, and computer science. From modeling projectile motion to optimizing economic models, quadratic equations provide valuable tools for analyzing and solving a wide range of problems. The ability to evaluate expressions like the one we have examined is therefore not just an academic skill but a practical asset with far-reaching implications. By mastering the techniques and concepts presented in this guide, readers will be well-equipped to tackle more complex mathematical challenges and appreciate the power and versatility of quadratic equations in various domains. The journey through this expression has not only provided a numerical solution but has also illuminated the beauty and interconnectedness of mathematical principles, highlighting the importance of a solid foundation in algebra for success in diverse fields of study and professional endeavors. The quadratic formula remains an indispensable tool in the mathematician's and scientist's toolkit, and a thorough understanding of its application is crucial for anyone seeking to engage with quantitative problem-solving in the modern world.