Solving The Quadratic Equation 2x² - 5x = -8 A Step-by-Step Guide
In the realm of mathematics, quadratic equations hold a prominent position, often encountered in various fields, from physics to engineering. Solving these equations is a fundamental skill, and this article delves into the process of finding the solution to the equation 2x² - 5x = -8. We will embark on a step-by-step journey, employing the renowned quadratic formula to unravel the roots of this equation. Along the way, we'll dissect the intricacies of complex numbers and their role in solutions, ensuring a comprehensive understanding of the concepts involved.
Transforming the Equation into Standard Form
Our initial step involves transforming the given equation, 2x² - 5x = -8, into the standard quadratic form, which is expressed as ax² + bx + c = 0. This transformation is crucial as it sets the stage for applying the quadratic formula. To achieve this, we simply add 8 to both sides of the equation, resulting in:
2x² - 5x + 8 = 0
Now, we have the equation neatly arranged in the standard quadratic form, where a = 2, b = -5, and c = 8. These coefficients will play a vital role in the subsequent application of the quadratic formula.
The Significance of Standard Form
The standard form of a quadratic equation is not merely a cosmetic rearrangement; it holds significant mathematical importance. This form allows us to readily identify the coefficients a, b, and c, which are essential components of the quadratic formula. Moreover, the standard form provides a clear visual representation of the equation's structure, making it easier to analyze and manipulate. It's the cornerstone for solving quadratic equations, enabling us to apply a consistent and reliable method.
Employing the Quadratic Formula: A Deep Dive
The quadratic formula stands as a cornerstone in the realm of algebra, providing a universal solution for quadratic equations of the form ax² + bx + c = 0. This formula elegantly expresses the roots of the equation in terms of its coefficients, offering a direct pathway to the solutions. The formula is stated as follows:
x = (-b ± √(b² - 4ac)) / (2a)
To decipher the roots of our specific equation, 2x² - 5x + 8 = 0, we substitute the values of a, b, and c, which we identified earlier as a = 2, b = -5, and c = 8, into the quadratic formula. This substitution sets the stage for the algebraic dance that will reveal the solutions.
Unveiling the Power of the Quadratic Formula
The quadratic formula's significance lies in its ability to solve any quadratic equation, regardless of its complexity. It's a versatile tool that transcends specific cases, providing a reliable method for finding the roots. The formula's structure also offers insights into the nature of the solutions. The discriminant, b² - 4ac, embedded within the formula, dictates whether the roots are real or complex, distinct or repeated. This inherent ability to analyze the solutions makes the quadratic formula an indispensable asset in mathematical problem-solving.
Delving into the Discriminant: Unveiling the Nature of Solutions
Within the quadratic formula, the expression b² - 4ac, known as the discriminant, plays a pivotal role in determining the nature of the solutions. The discriminant acts as a mathematical oracle, revealing whether the roots are real or complex, distinct or repeated. Its value provides crucial insights into the behavior of the quadratic equation.
In our case, where a = 2, b = -5, and c = 8, the discriminant is calculated as follows:
b² - 4ac = (-5)² - 4 * 2 * 8 = 25 - 64 = -39
The discriminant, -39, is a negative number. This negative value signifies that the solutions to the quadratic equation are complex numbers. Complex numbers, which involve the imaginary unit 'i' (where i² = -1), arise when the discriminant is negative, indicating that the square root of a negative number is involved in the solution.
The Discriminant's Tale: Decoding the Solution Landscape
The discriminant's value acts as a key to unlocking the secrets of a quadratic equation's solutions. A positive discriminant signals the presence of two distinct real roots, indicating that the equation intersects the x-axis at two different points. A zero discriminant signifies a single, repeated real root, implying that the equation touches the x-axis at one point. Conversely, a negative discriminant, as we've encountered, unveils the existence of complex roots, indicating that the equation does not intersect the x-axis in the real number plane. This ability to discern the nature of solutions makes the discriminant a powerful tool in the analysis of quadratic equations.
Substituting and Simplifying: The Path to the Solutions
With the discriminant calculated and its implications understood, we now embark on the crucial step of substituting the values of a, b, and the discriminant back into the quadratic formula. This substitution sets the stage for the final algebraic simplification that will reveal the solutions to our equation.
Recall the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
Substituting a = 2, b = -5, and the discriminant (b² - 4ac) = -39, we get:
x = (-(-5) ± √(-39)) / (2 * 2)
Simplifying the expression, we arrive at:
x = (5 ± √(-39)) / 4
Now, we encounter the square root of a negative number, √(-39). This is where the realm of complex numbers comes into play. We can express √(-39) as √(-1 * 39) = √(-1) * √(39) = i√(39), where 'i' is the imaginary unit.
Therefore, the solutions become:
x = (5 ± i√(39)) / 4
The Art of Simplification: Unveiling the Solutions' Elegance
Simplification is not merely about reducing an expression to its most concise form; it's about revealing the underlying structure and elegance of the solution. In the context of the quadratic formula, simplification involves carefully handling the square root, dealing with negative signs, and expressing the solutions in their most understandable form. Each step of simplification brings us closer to the final answer, unveiling the roots of the equation in their full clarity. This process highlights the beauty of mathematical manipulation, where seemingly complex expressions are transformed into elegant solutions.
Expressing the Solutions: Embracing Complex Numbers
The solutions we've obtained, x = (5 ± i√(39)) / 4, are complex numbers. Complex numbers, denoted in the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit, extend the realm of numbers beyond the familiar real number line. They are essential in various mathematical and scientific domains, including electrical engineering, quantum mechanics, and signal processing.
To express our solutions explicitly, we separate the real and imaginary parts:
x = 5/4 ± (i√(39))/4
Thus, the two solutions are:
x₁ = 5/4 + (i√(39))/4 x₂ = 5/4 - (i√(39))/4
These solutions represent two distinct points in the complex plane, a two-dimensional space where the horizontal axis represents the real part and the vertical axis represents the imaginary part.
The Embrace of Complex Numbers: Expanding the Mathematical Horizon
Complex numbers, often initially perceived as abstract entities, are in fact powerful tools that expand the scope of mathematical solutions. They allow us to solve equations that have no real solutions, providing a complete picture of the solution landscape. Their applications extend far beyond pure mathematics, permeating various scientific and engineering disciplines. Embracing complex numbers enriches our mathematical understanding and equips us to tackle a broader range of problems.
Identifying the Correct Option: A Triumphant Conclusion
Having meticulously navigated the steps of solving the quadratic equation, we arrive at the solutions:
x = (5 ± i√(39)) / 4
Now, we compare our solutions with the options provided:
A. (-5 ± i√(39)) / 4 B. (5 ± i√(39)) / 4 C. (5 ± i√(39)) / 2 D. (-5 ± i√(192)) / 2
By careful comparison, we can confidently identify that option B, (5 ± i√(39)) / 4, precisely matches the solutions we have derived. Therefore, option B is the correct answer.
The Triumph of Solution: A Journey's End
Reaching the correct solution is not merely the end of a mathematical exercise; it's the culmination of a journey through logical reasoning, algebraic manipulation, and conceptual understanding. Each step, from transforming the equation to applying the quadratic formula and simplifying the results, has contributed to this triumphant conclusion. The satisfaction of arriving at the correct answer is a testament to the power of mathematical problem-solving and the elegance of the underlying principles.
In conclusion, the solution to the quadratic equation 2x² - 5x = -8 is indeed (5 ± i√(39)) / 4, as confirmed by our step-by-step analysis. This exploration has not only provided the answer but also illuminated the process of solving quadratic equations, emphasizing the significance of the quadratic formula, the role of the discriminant, and the embrace of complex numbers. Armed with this knowledge, we are better equipped to tackle future mathematical challenges.
