Solving 2r² + 3r + 14 = 8 A Step-by-Step Guide

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In the realm of mathematics, quadratic equations hold a prominent position, serving as fundamental tools for modeling diverse phenomena across various disciplines. These equations, characterized by their second-degree polynomial form, often present a captivating challenge to solve. In this comprehensive guide, we delve into the intricacies of solving the specific quadratic equation 2r² + 3r + 14 = 8, meticulously exploring the underlying concepts and techniques involved.

H2: Understanding Quadratic Equations

At the heart of our exploration lies the quadratic equation, a mathematical expression that takes the general form of ax² + bx + c = 0, where 'a', 'b', and 'c' represent constant coefficients, and 'x' denotes the variable we seek to determine. The solutions to a quadratic equation, also known as roots, are the values of 'x' that satisfy the equation, effectively making the expression equal to zero. These roots hold significant importance, as they often represent critical points or values within the context of the problem being modeled.

Before we embark on the journey of solving our specific equation, let's solidify our understanding of the key concepts that underpin quadratic equations. The coefficients 'a', 'b', and 'c' play a crucial role in shaping the behavior and characteristics of the quadratic equation. The coefficient 'a' determines the parabola's concavity, whether it opens upwards or downwards. The coefficient 'b' influences the position of the parabola's axis of symmetry, while the coefficient 'c' represents the y-intercept, the point where the parabola intersects the vertical axis. The discriminant, a pivotal element in analyzing quadratic equations, is calculated as b² - 4ac. This value provides invaluable insights into the nature of the roots. A positive discriminant signifies two distinct real roots, indicating that the parabola intersects the x-axis at two distinct points. A zero discriminant implies a single real root, where the parabola touches the x-axis at only one point. Conversely, a negative discriminant unveils the presence of two complex roots, implying that the parabola does not intersect the x-axis in the real number plane.

H2: Transforming the Equation into Standard Form

Our initial equation, 2r² + 3r + 14 = 8, stands poised for transformation into the standard quadratic form, ax² + bx + c = 0. This crucial step streamlines the solving process, allowing us to readily apply established techniques. To achieve this transformation, we embark on a mission to eliminate the constant term on the right-hand side of the equation. Our weapon of choice is the principle of equality, which dictates that performing the same operation on both sides of an equation preserves its balance. By subtracting 8 from both sides of the equation, we effectively neutralize the constant term on the right-hand side, paving the way for the standard form. This maneuver gracefully transforms the equation into 2r² + 3r + 6 = 0, a form that aligns perfectly with the quadratic equation paradigm.

This strategic transformation marks a pivotal moment in our quest to solve the equation. With the equation now nestled snugly in the standard form, we gain a clear perspective on the coefficients that govern its behavior. The coefficient 'a' stands proudly at 2, indicating the parabola's upward concavity. The coefficient 'b' settles at 3, influencing the parabola's axis of symmetry. And the coefficient 'c' rests at 6, marking the y-intercept. These coefficients serve as vital ingredients in our upcoming steps, guiding us towards the solutions that satisfy the equation.

H2: Employing the Quadratic Formula

The quadratic formula emerges as a potent ally in our quest to conquer the equation 2r² + 3r + 6 = 0. This formula, a timeless masterpiece of mathematical ingenuity, provides a universal solution for quadratic equations, gracefully handling both real and complex roots. The quadratic formula, expressed as

r = (-b ± √(b² - 4ac)) / 2a

weaves together the coefficients 'a', 'b', and 'c' in a harmonious dance, revealing the hidden roots that lie within the equation. To harness the power of the quadratic formula, we must first identify the coefficients that govern our equation. In this case, we have a = 2, b = 3, and c = 6. These values serve as the keys that unlock the solutions to our equation.

With the coefficients firmly in hand, we embark on the journey of substituting them into the quadratic formula. Each coefficient finds its designated place, weaving a tapestry of numbers that will lead us to the roots. Substituting these values into the formula, we obtain:

r = (-3 ± √(3² - 4 * 2 * 6)) / (2 * 2)

This expression, a symphony of numbers and operations, encapsulates the essence of the quadratic formula. Now, the stage is set for simplification, a meticulous process that will unveil the hidden solutions. We embark on this simplification journey with unwavering determination, peeling away the layers of complexity to reveal the roots that lie within.

H2: Simplifying the Expression

Our expression, r = (-3 ± √(3² - 4 * 2 * 6)) / (2 * 2), now stands as a testament to our progress. It's time to embark on a journey of simplification, where we meticulously unravel the mathematical knots to reveal the solutions that lie hidden within. We begin our simplification quest by focusing on the expression nestled under the square root, the very heart of the quadratic formula. We perform the operations within the square root with precision, following the order of operations to ensure accuracy.

First, we square the value of 3, obtaining 9. Then, we multiply 4, 2, and 6, yielding 48. Next, we subtract 48 from 9, resulting in -39. This negative value under the square root hints at the presence of complex roots, a fascinating twist in our mathematical journey. Our expression now transforms into:

r = (-3 ± √(-39)) / 4

The negative sign under the square root compels us to venture into the realm of imaginary numbers, where the symbol 'i' represents the square root of -1. We rewrite √(-39) as √(39 * -1), which then elegantly transforms into √39 * √-1, or √39 * i. This step unveils the imaginary component of our solutions, adding a layer of intrigue to our quest.

With the imaginary component revealed, our expression takes on a new form:

r = (-3 ± i√39) / 4

This expression, now a harmonious blend of real and imaginary components, represents the solutions to our quadratic equation. We have successfully navigated the complexities of the quadratic formula and arrived at the simplified form, ready to unveil the individual roots.

H2: Identifying the Solutions

Our expression, r = (-3 ± i√39) / 4, now stands as a beacon, illuminating the path to the solutions of our quadratic equation. The '±' symbol signifies a fork in the road, leading us to two distinct solutions, each with its own unique character. To unveil these solutions, we must carefully consider both the positive and negative scenarios.

First, let's embrace the positive scenario, where the '±' symbol transforms into a '+'. This path leads us to the first solution:

r₁ = (-3 + i√39) / 4

This solution, a harmonious blend of real and imaginary components, represents one of the roots that satisfy our quadratic equation. The real part, -3/4, anchors the solution on the number line, while the imaginary part, (√39)/4 * i, extends it into the complex plane. Now, let's turn our attention to the negative scenario, where the '±' symbol transforms into a '-'. This path unveils the second solution:

r₂ = (-3 - i√39) / 4

This solution, a mirror image of the first, also possesses both real and imaginary components. The real part remains at -3/4, while the imaginary part takes on the opposite sign, becoming -(√39)/4 * i. These two solutions, r₁ and r₂, stand as the culmination of our mathematical journey. They are the values of 'r' that, when plugged back into the original equation, make the expression equal to zero. They represent the points where the parabola intersects the complex plane, a testament to the power and elegance of quadratic equations.

H2: Conclusion

In this comprehensive exploration, we embarked on a journey to solve the quadratic equation 2r² + 3r + 14 = 8. We began by understanding the fundamental concepts of quadratic equations, delving into the significance of coefficients and the discriminant. We then transformed the equation into standard form, preparing it for the application of the quadratic formula. With the quadratic formula as our guide, we meticulously simplified the expression, navigating the realm of imaginary numbers and unveiling the complex roots. Finally, we identified the two distinct solutions:

r₁ = (-3 + i√39) / 4 r₂ = (-3 - i√39) / 4

These solutions, a testament to the power of mathematical techniques, represent the culmination of our efforts. We have successfully solved the equation, gaining a deeper appreciation for the beauty and intricacies of quadratic equations.

This journey serves as a reminder that mathematics is not merely a collection of formulas and procedures; it is a powerful tool for understanding and modeling the world around us. By mastering the techniques for solving quadratic equations, we equip ourselves with a valuable skill that can be applied across various disciplines, from physics and engineering to economics and finance. The ability to solve quadratic equations opens doors to a world of possibilities, empowering us to tackle complex problems and gain deeper insights into the intricate patterns that govern our universe.

  • The quadratic formula is a universal tool for solving quadratic equations, handling both real and complex roots.
  • The discriminant (b² - 4ac) provides valuable insights into the nature of the roots.
  • Complex roots arise when the discriminant is negative, indicating that the parabola does not intersect the x-axis in the real number plane.
  • Solving quadratic equations is a fundamental skill with applications across various disciplines.