Solve The Quadratic Equation 5-2x²=6x
In the realm of mathematics, quadratic equations hold a prominent position, serving as the foundation for countless applications in science, engineering, and beyond. These equations, characterized by their highest power of the variable being two, often present a captivating challenge to problem-solvers. In this comprehensive guide, we embark on a journey to dissect and conquer the quadratic equation 5-2x²=6x, unraveling its intricacies and equipping you with the tools to confidently tackle similar problems. Our discussion will delve into the fundamental concepts, step-by-step solutions, and the underlying principles that govern these mathematical expressions. We will explore the standard form of quadratic equations, the significance of the coefficients, and the various methods employed to extract the elusive roots – the values of the variable that satisfy the equation. By the end of this exploration, you will not only possess the ability to solve this specific equation but also gain a profound understanding of the broader landscape of quadratic equations.
Understanding the Quadratic Equation
Before diving into the solution, let's establish a solid understanding of what constitutes a quadratic equation. In its standard form, a quadratic equation is expressed as ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' represents the variable we seek to determine. The coefficient 'a' plays a crucial role, dictating the concavity of the parabola represented by the equation. A non-zero 'a' value ensures the equation remains quadratic, while 'b' and 'c' influence the position and intersection points of the parabola with the x-axis. These intersection points, also known as roots or solutions, hold the key to unlocking the equation's secrets. Finding these roots involves a blend of algebraic manipulation, strategic application of formulas, and a keen eye for patterns. Whether you're a student grappling with algebraic concepts or a seasoned mathematician seeking to refine your skills, a thorough understanding of quadratic equations is an indispensable asset.
Transforming the Equation: Standard Form
Our initial equation, 5-2x²=6x, may appear slightly out of sync with the standard form we've just discussed. However, with a few algebraic maneuvers, we can seamlessly transform it into the familiar ax² + bx + c = 0 structure. The key lies in rearranging the terms, ensuring all components reside on one side of the equation, leaving zero on the other. To achieve this, we embark on a systematic journey, carefully adding or subtracting terms from both sides while preserving the equation's delicate balance. First, we add 2x² to both sides, effectively eliminating the negative term on the left. Next, we subtract 5 from both sides, isolating the variable terms. The result of this meticulous dance is a transformed equation: 2x² + 6x - 5 = 0. This newly acquired form aligns perfectly with the standard quadratic equation, allowing us to readily identify the coefficients 'a', 'b', and 'c'. In this case, 'a' assumes the value of 2, 'b' proudly stands at 6, and 'c' contributes -5 to the equation's character. With the equation now in its standard form, we are primed to unleash the arsenal of techniques designed to extract its hidden roots.
Methods for Solving Quadratic Equations
With our equation neatly arranged in standard form, we are now poised to explore the diverse landscape of solution methods. The realm of quadratic equations offers a rich tapestry of approaches, each with its own strengths and nuances. Among the most prominent techniques are factoring, completing the square, and the quadratic formula. Factoring, a method rooted in the art of decomposition, seeks to express the quadratic expression as a product of two linear factors. When successful, factoring elegantly reveals the roots as the values that make each factor equal to zero. Completing the square, a technique championed for its versatility, transforms the equation into a perfect square trinomial, allowing us to isolate the variable through strategic manipulation. The quadratic formula, a true workhorse in the world of quadratic equations, provides a universal solution, applicable to any equation regardless of its factorability. This formula, derived from the method of completing the square, directly calculates the roots based on the coefficients 'a', 'b', and 'c'. The choice of method often depends on the specific characteristics of the equation, personal preference, and the desired level of elegance in the solution. In our quest to solve 2x² + 6x - 5 = 0, we will delve into the power of the quadratic formula, a reliable companion in the pursuit of roots.
Applying the Quadratic Formula
The quadratic formula, a cornerstone of quadratic equation solving, stands as a testament to mathematical ingenuity. This formula, expressed as x = (-b ± √(b² - 4ac)) / 2a, provides a direct pathway to the roots of any quadratic equation in standard form. The symbols within the formula hold profound significance, each representing a key element of the equation's structure. 'a', 'b', and 'c', as we've previously encountered, are the coefficients that define the quadratic equation. The ± symbol signifies the existence of two potential roots, one obtained by addition and the other by subtraction. The square root term, √(b² - 4ac), known as the discriminant, plays a pivotal role in determining the nature of the roots. A positive discriminant heralds two distinct real roots, a zero discriminant signals a single real root (a repeated root), and a negative discriminant unveils a pair of complex roots. To apply the quadratic formula to our equation, 2x² + 6x - 5 = 0, we meticulously substitute the values of 'a', 'b', and 'c'. 'a' assumes the value of 2, 'b' proudly stands at 6, and 'c' contributes -5. Plugging these values into the formula, we embark on a journey of simplification, carefully navigating the arithmetic operations to reveal the equation's hidden roots. This process, while seemingly mechanical, underscores the elegance and power of mathematical formulas in unraveling complex problems.
Calculating the Discriminant
Before we fully unleash the power of the quadratic formula, let's take a detour to explore the significance of the discriminant, the enigmatic expression nestled beneath the square root sign: b² - 4ac. The discriminant acts as a mathematical divining rod, offering insights into the nature of the roots even before we embark on the full calculation. As we briefly touched upon earlier, the discriminant's value dictates whether the equation boasts two distinct real roots, a single repeated real root, or a pair of complex roots. A positive discriminant, a beacon of distinct real roots, signals that the parabola represented by the equation intersects the x-axis at two distinct points. A zero discriminant, a harbinger of a single repeated real root, indicates that the parabola grazes the x-axis at exactly one point, the vertex of the parabola. A negative discriminant, a veil concealing complex roots, reveals that the parabola never crosses the x-axis, its roots residing in the realm of complex numbers. For our equation, 2x² + 6x - 5 = 0, let's calculate the discriminant. Substituting the values of 'a', 'b', and 'c', we have 6² - 4 * 2 * (-5), which simplifies to 36 + 40, resulting in a discriminant of 76. This positive value assures us that our equation possesses two distinct real roots, setting the stage for the final act of root extraction.
Determining the Roots
With the discriminant's revelation of two distinct real roots, we now return to the quadratic formula, armed with confidence and anticipation. We have already laid the groundwork by substituting the values of 'a', 'b', and 'c' into the formula. Now, we embark on the final leg of our journey, simplifying the expression to unveil the precise values of the roots. The formula, x = (-b ± √(b² - 4ac)) / 2a, transforms into x = (-6 ± √76) / (2 * 2) upon substitution. The square root of 76, while not a perfect square, can be simplified by factoring out the largest perfect square factor, which is 4. √76 then becomes √(4 * 19), which simplifies to 2√19. Our equation now stands as x = (-6 ± 2√19) / 4. To further refine our solution, we can divide both the numerator and denominator by 2, resulting in x = (-3 ± √19) / 2. This elegant expression represents our two roots, one obtained by addition and the other by subtraction. Therefore, the roots of the quadratic equation 2x² + 6x - 5 = 0 are x = (-3 + √19) / 2 and x = (-3 - √19) / 2. These values, though seemingly abstract, hold the key to understanding the equation's behavior and its intersections with the x-axis.
Expressing the Solutions
Having meticulously calculated the roots, our final task is to express them in a clear and concise manner. The roots, x = (-3 + √19) / 2 and x = (-3 - √19) / 2, represent the solutions to our quadratic equation, the values of 'x' that satisfy the equation when plugged in. While these expressions are mathematically accurate, they can be further refined for clarity and practical application. We can choose to leave the roots in their exact form, as we have them now, preserving the precision of the solution. Alternatively, we can approximate the roots by calculating the decimal values of √19 and performing the arithmetic operations. √19 is approximately 4.36, so the roots become approximately x = (-3 + 4.36) / 2 and x = (-3 - 4.36) / 2. Simplifying these expressions, we obtain x ≈ 0.68 and x ≈ -3.68. These approximate values provide a tangible sense of the roots' location on the number line. Whether we opt for exact or approximate values, the key is to communicate the solutions clearly and in a manner that suits the context of the problem. In many practical applications, approximate values are preferred for ease of interpretation and use.
Conclusion: Mastering Quadratic Equations
Our journey through the quadratic equation 5-2x²=6x has been a testament to the power of mathematical principles and techniques. We embarked on this quest by transforming the equation into its standard form, unlocking the gateway to a diverse array of solution methods. We then delved into the elegance and universality of the quadratic formula, a tool that empowers us to conquer any quadratic equation. Along the way, we encountered the discriminant, a mathematical oracle that unveiled the nature of the roots before we even embarked on the full calculation. We meticulously applied the quadratic formula, simplified the expressions, and extracted the precise values of the roots. Finally, we expressed the solutions in a clear and concise manner, ready for practical application. This journey not only equips you with the ability to solve this specific equation but also instills a deep understanding of quadratic equations in general. You now possess the knowledge to confidently tackle similar problems, armed with the tools of algebraic manipulation, strategic formula application, and a keen eye for mathematical patterns. The world of quadratic equations, once a daunting challenge, now stands before you as a realm of solvable mysteries, ready to be explored and conquered.
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