Analyzing The Quadratic Equation 3x² - 8x + 5 = 5x² And Its Roots
At the heart of algebra lies the quadratic equation, a powerful tool for modeling and solving a wide range of problems. Quadratic equations are defined as polynomial equations of the second degree, meaning they contain at least one term that is squared. The standard form of a quadratic equation is expressed as ax² + bx + c = 0, where a, b, and c are constants, and x represents the variable we aim to solve for. These equations manifest in diverse scenarios, from calculating projectile trajectories in physics to optimizing geometric shapes in engineering. To effectively tackle quadratic equations, one must grasp key concepts such as the discriminant, roots, and various solution methods.
One common method for solving quadratic equations involves factoring, which entails breaking down the quadratic expression into the product of two linear factors. Another widely used technique is the quadratic formula, a universal solution that directly yields the roots of the equation regardless of its factorability. The discriminant, denoted as Δ = b² - 4ac, plays a crucial role in determining the nature and number of roots a quadratic equation possesses. It provides valuable insights into whether the equation has two distinct real roots, one repeated real root, or two complex roots. A positive discriminant indicates two distinct real roots, a zero discriminant implies one repeated real root, and a negative discriminant signifies two complex roots. Mastering these concepts and techniques empowers individuals to confidently solve quadratic equations and apply them to real-world situations.
In this article, we will delve into a detailed analysis of the specific quadratic equation 3x² - 8x + 5 = 5x², exploring its properties, discriminant, and the nature of its roots. By meticulously examining each aspect of the equation, we aim to provide a comprehensive understanding of how to approach and solve quadratic equations effectively. Whether you're a student grappling with algebraic concepts or a professional seeking to refine your problem-solving skills, this guide will equip you with the knowledge and confidence to tackle quadratic equations with precision and clarity. Understanding the nuances of quadratic equations is not merely an academic exercise; it is a gateway to unlocking the power of mathematical modeling and problem-solving in numerous fields. So, let's embark on this journey to unravel the intricacies of the quadratic equation and discover its profound applications.
Problem Statement: Analyzing the Equation 3x² - 8x + 5 = 5x²
The specific quadratic equation we will be dissecting is 3x² - 8x + 5 = 5x². This equation, while seemingly straightforward, holds a wealth of information waiting to be unveiled. Our primary objective is to determine which statement about this equation is true, focusing particularly on the discriminant and the nature of its roots. The discriminant, as mentioned earlier, is a critical component in understanding the behavior of quadratic equations, and its value dictates whether the equation has real or complex roots, and how many of each. To accurately assess the given statements, we must first manipulate the equation into its standard form, which is ax² + bx + c = 0. This standardization is a crucial first step as it allows us to readily identify the coefficients a, b, and c, which are essential for calculating the discriminant.
By rearranging the terms of the equation, we can bring it into the familiar standard form, making it easier to apply the tools and techniques of quadratic equation analysis. Once in standard form, we can then proceed to calculate the discriminant using the formula Δ = b² - 4ac. This calculation will provide us with a numerical value that holds the key to understanding the roots of the equation. A positive discriminant signals that the equation has two distinct real roots, indicating that there are two unique solutions for x that satisfy the equation. A discriminant of zero implies that the equation has one repeated real root, meaning there is only one solution for x. Conversely, a negative discriminant indicates that the equation has two complex roots, which are solutions that involve imaginary numbers. By carefully interpreting the value of the discriminant, we can definitively determine the nature of the roots and select the correct statement about the equation.
This problem is not just an exercise in algebraic manipulation; it is a journey into the heart of mathematical problem-solving. It requires a systematic approach, attention to detail, and a firm grasp of fundamental concepts. By working through this problem, we not only reinforce our understanding of quadratic equations but also develop critical thinking skills that are applicable across a wide range of disciplines. So, let's proceed with confidence, knowing that each step we take brings us closer to unraveling the mysteries of this equation and gaining a deeper appreciation for the elegance and power of mathematics. The process of solving this equation is a testament to the beauty of mathematical reasoning and its ability to illuminate the world around us.
Solution: Step-by-Step Breakdown
Step 1: Rearrange the Equation
The initial step in solving this problem is to rearrange the quadratic equation 3x² - 8x + 5 = 5x² into the standard form of a quadratic equation, which is ax² + bx + c = 0. This form is essential because it allows us to easily identify the coefficients a, b, and c, which are necessary for calculating the discriminant. To achieve this, we need to subtract 5x² from both sides of the equation. This operation will consolidate all the terms on one side, leaving zero on the other side, thus aligning the equation with the standard form.
Subtracting 5x² from both sides gives us: 3x² - 8x + 5 - 5x² = 5x² - 5x². Simplifying this, we get -2x² - 8x + 5 = 0. Now, to make the leading coefficient positive, we can multiply the entire equation by -1. This gives us 2x² + 8x - 5 = 0. This form of the equation is now in the standard quadratic form, where a = 2, b = 8, and c = -5. Identifying these coefficients is a crucial step as they are the building blocks for calculating the discriminant and subsequently determining the nature of the roots of the equation. This rearrangement not only simplifies the equation but also sets the stage for the next steps in our problem-solving journey. The importance of this step cannot be overstated, as it transforms the original equation into a more manageable and recognizable form, paving the way for further analysis.
Step 2: Calculate the Discriminant
Now that we have the quadratic equation in standard form (2x² + 8x - 5 = 0), we can proceed to calculate the discriminant. The discriminant, denoted as Δ, is a critical value that provides information about the nature of the roots of the quadratic equation. The formula for the discriminant is Δ = b² - 4ac, where a, b, and c are the coefficients of the quadratic equation in the standard form. In our equation, a = 2, b = 8, and c = -5. Substituting these values into the discriminant formula, we get Δ = 8² - 4(2)(-5).
Simplifying this expression, we first calculate 8², which is 64. Then, we calculate 4(2)(-5), which equals -40. Substituting these values back into the equation, we have Δ = 64 - (-40). Further simplification gives us Δ = 64 + 40, which results in Δ = 104. The discriminant, in this case, is 104. This positive value of the discriminant is a significant indicator, as it tells us that the quadratic equation has two distinct real roots. The magnitude of the discriminant also provides insights into the separation of these roots on the number line, but for our primary objective of determining the nature of the roots, the positive sign is the key piece of information. This calculation is a pivotal moment in our analysis, as it directly addresses the core question of the problem and sets the foundation for our final conclusion.
Step 3: Determine the Nature of the Roots
Having calculated the discriminant (Δ) as 104, we can now definitively determine the nature of the roots of the quadratic equation 2x² + 8x - 5 = 0. The discriminant serves as a crucial indicator in this regard. As a general rule, if the discriminant is greater than 0 (Δ > 0), the quadratic equation has two distinct real roots. This means that there are two unique values of x that will satisfy the equation. If the discriminant is equal to 0 (Δ = 0), the quadratic equation has one repeated real root, indicating that there is only one value of x that satisfies the equation. Finally, if the discriminant is less than 0 (Δ < 0), the quadratic equation has two complex roots, which involve imaginary numbers and are not real numbers.
In our case, the discriminant is 104, which is clearly greater than 0. Therefore, based on the rules governing the discriminant, we can conclude that the quadratic equation 2x² + 8x - 5 = 0 has two distinct real roots. This conclusion directly answers the core question posed in the problem statement and allows us to confidently select the correct statement from the given options. The positive value of the discriminant acts as a definitive signpost, guiding us to the understanding that the equation has two real solutions. This step not only provides the answer but also reinforces the fundamental relationship between the discriminant and the nature of the roots of a quadratic equation. The ability to interpret the discriminant is a cornerstone of quadratic equation analysis, and this step exemplifies its importance in problem-solving.
Conclusion: Identifying the Correct Statement
Based on our step-by-step analysis, we have successfully determined that the discriminant of the quadratic equation 3x² - 8x + 5 = 5x² is 104, which is greater than 0. This crucial finding allows us to conclude that the equation has two distinct real roots. Now, we can confidently identify the correct statement among the options provided. The correct statement is: B. The discriminant is greater than 0, so there are two real roots.
This conclusion is a direct result of our methodical approach, starting with rearranging the equation into standard form, calculating the discriminant, and then interpreting its value. Each step was essential in leading us to the correct answer. The discriminant, in this case, acted as a powerful tool, providing us with a clear and concise way to understand the nature of the roots of the equation. This problem exemplifies the importance of understanding the fundamental concepts of quadratic equations, such as the discriminant, and how they can be applied to solve real-world problems. The ability to analyze quadratic equations and interpret their properties is a valuable skill in various fields, from mathematics and physics to engineering and finance. By mastering these concepts, individuals can tackle complex problems with confidence and precision.
Furthermore, this exercise highlights the importance of a systematic approach to problem-solving. By breaking down the problem into smaller, manageable steps, we were able to navigate the complexities of the equation and arrive at a clear and accurate conclusion. This approach is not only effective for solving mathematical problems but also applicable to a wide range of challenges in various domains. The journey of solving this quadratic equation has been a testament to the power of mathematical reasoning and the importance of a methodical approach in problem-solving. We hope that this detailed analysis has provided a comprehensive understanding of quadratic equations and their applications, empowering you to tackle similar problems with confidence and clarity.
