Discriminant Of 3x² - 6x + 5 Calculation And Explanation

In the realm of mathematics, particularly when dealing with quadratic equations, the discriminant plays a pivotal role in determining the nature of the roots of a quadratic equation. The discriminant, often denoted as Δ, is a crucial component derived from the coefficients of the quadratic equation. In this article, we will delve into the trinomial equation $3x^2 - 6x + 5$, and we aim to unravel the value of its discriminant. Understanding the discriminant is essential as it provides valuable insights into whether the quadratic equation has real roots, distinct real roots, or complex roots. We will explore the formula for calculating the discriminant and apply it to the given trinomial. By the end of this discussion, you will have a comprehensive understanding of how to compute the discriminant and interpret its significance in the context of quadratic equations. This knowledge is not only beneficial for academic purposes but also for various applications in engineering, physics, and computer science, where quadratic equations frequently arise. The discriminant is a fundamental concept that bridges the gap between the algebraic representation of a quadratic equation and the geometric interpretation of its roots on the complex plane. So, let's embark on this mathematical journey to demystify the discriminant and its practical implications.

What is the Discriminant?

To truly grasp the essence of the discriminant, one must first understand its significance within the broader context of quadratic equations. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form $ax^2 + bx + c = 0$, where 'a', 'b', and 'c' are coefficients, and 'x' is the variable. The discriminant, denoted as Δ, is a mathematical expression derived from these coefficients, specifically calculated as $Δ = b^2 - 4ac$. This seemingly simple formula holds the key to unlocking the nature of the roots of the quadratic equation. The roots, also known as solutions, are the values of 'x' that satisfy the equation. The discriminant acts as a powerful indicator, revealing whether these roots are real and distinct, real and equal, or complex conjugates. When the discriminant is positive ($Δ > 0$), the quadratic equation has two distinct real roots, indicating that the parabola represented by the equation intersects the x-axis at two different points. If the discriminant is zero ($Δ = 0$), the equation has exactly one real root (or two equal real roots), meaning the parabola touches the x-axis at a single point. Conversely, when the discriminant is negative ($Δ < 0$), the equation has no real roots, but instead, it has two complex conjugate roots, signifying that the parabola does not intersect the x-axis. The discriminant, therefore, is not just a mere calculation; it is a critical tool for analyzing the behavior and characteristics of quadratic equations, providing a concise and elegant way to understand the nature of their solutions without explicitly solving for them. Its applications extend beyond pure mathematics, playing a crucial role in various fields where quadratic equations are employed to model real-world phenomena.

Calculating the Discriminant for 3x² - 6x + 5

In this section, we will focus on the practical application of the discriminant formula to the given trinomial, $3x^2 - 6x + 5$. The process involves identifying the coefficients 'a', 'b', and 'c' from the trinomial and then substituting these values into the discriminant formula, $Δ = b^2 - 4ac$. This step-by-step calculation will demonstrate how the discriminant is computed in a concrete example, making the concept more tangible and easier to understand. The trinomial $3x^2 - 6x + 5$ is a perfect example of a quadratic equation, where the coefficients are clearly defined. Here, 'a' is the coefficient of the $x^2$ term, which is 3; 'b' is the coefficient of the x term, which is -6; and 'c' is the constant term, which is 5. Once these coefficients are identified, the next step is to meticulously substitute them into the discriminant formula. This involves replacing 'b' with -6, 'a' with 3, and 'c' with 5 in the equation $Δ = b^2 - 4ac$. The calculation then proceeds as follows: $Δ = (-6)^2 - 4 * 3 * 5$. It is crucial to follow the order of operations (PEMDAS/BODMAS) to ensure the correct result. First, we square -6, which gives us 36. Then, we multiply 4, 3, and 5, which results in 60. Finally, we subtract 60 from 36, which yields the discriminant value. This methodical approach to calculating the discriminant not only provides the numerical answer but also reinforces the understanding of how each coefficient contributes to the final result. By walking through this example, readers can gain confidence in their ability to calculate the discriminant for any given quadratic equation, paving the way for further analysis of the equation's roots.

Step-by-Step Calculation

Let's break down the calculation of the discriminant for the trinomial $3x^2 - 6x + 5$ into a clear, step-by-step process. This detailed walkthrough will ensure that the method is transparent and easy to follow, even for those who are new to the concept. By meticulously examining each step, we can avoid common errors and reinforce the correct application of the discriminant formula. First, we identify the coefficients in the trinomial. As established earlier, the coefficient 'a' is 3, 'b' is -6, and 'c' is 5. These values are the foundation of our calculation, and accurately identifying them is crucial for obtaining the correct discriminant value. The next step involves substituting these values into the discriminant formula, $Δ = b^2 - 4ac$. Replacing 'b' with -6, 'a' with 3, and 'c' with 5, we get $Δ = (-6)^2 - 4 * 3 * 5$. Now, we proceed with the arithmetic operations, adhering to the order of operations (PEMDAS/BODMAS). We begin by squaring -6, which results in 36. This is a critical step, as squaring a negative number yields a positive result. Next, we perform the multiplication: 4 multiplied by 3 multiplied by 5 equals 60. This part of the calculation represents the product of the coefficients and the constant factor in the discriminant formula. Finally, we subtract the result of the multiplication from the squared term: 36 - 60. This subtraction gives us the final value of the discriminant. Performing the subtraction, we find that 36 - 60 equals -24. Therefore, the discriminant, Δ, for the trinomial $3x^2 - 6x + 5$ is -24. This step-by-step calculation not only provides the answer but also elucidates the process, making it easier to replicate and apply to other quadratic equations. The result, -24, carries significant information about the nature of the roots of the trinomial, which we will explore in the next section.

Result and Interpretation

Having meticulously calculated the discriminant for the trinomial $3x^2 - 6x + 5$, we have arrived at the result: Δ = -24. Now, the crucial step is to interpret what this value signifies about the nature of the roots of the quadratic equation. The discriminant, as we've learned, is a powerful indicator that provides insights into whether the roots are real, distinct, equal, or complex. In this case, the negative value of the discriminant, -24, immediately tells us that the quadratic equation has no real roots. This is a fundamental concept: a negative discriminant implies that the parabola represented by the equation does not intersect the x-axis. Instead, the roots are complex conjugates, meaning they involve the imaginary unit 'i', where $i^2 = -1$. Complex roots occur in pairs of the form a + bi and a - bi, where 'a' and 'b' are real numbers. These roots are not located on the real number line but exist in the complex plane. This interpretation is significant because it provides a complete picture of the solutions to the quadratic equation without actually solving for the roots explicitly. Knowing that the roots are complex conjugates allows us to understand the behavior of the quadratic function and its graph. For instance, when plotting the graph of $y = 3x^2 - 6x + 5$, we would observe that the parabola does not cross the x-axis, confirming the absence of real roots. The vertex of the parabola, however, would still be a significant point, representing either the minimum or maximum value of the function, depending on the sign of the leading coefficient 'a'. In summary, a negative discriminant, such as -24 in our example, is a clear signal that the quadratic equation has complex conjugate roots and no real roots. This understanding is vital for various applications, from solving mathematical problems to modeling physical systems where complex solutions might arise.

Conclusion

In conclusion, the discriminant of a quadratic equation serves as a cornerstone in understanding the nature of its roots. For the trinomial $3x^2 - 6x + 5$, we meticulously calculated the discriminant using the formula $Δ = b^2 - 4ac$, and we arrived at the value of -24. This negative value is not just a numerical result; it is a profound indicator that the given quadratic equation possesses complex conjugate roots and lacks any real roots. Throughout this exploration, we've emphasized the step-by-step process of identifying the coefficients, substituting them into the formula, and performing the arithmetic operations with precision. This methodical approach ensures accuracy and reinforces the understanding of how each component of the equation contributes to the discriminant value. Furthermore, we've delved into the interpretation of the discriminant, highlighting its significance in determining the type of roots a quadratic equation has. A negative discriminant, as seen in our example, signifies the presence of complex roots, while a positive discriminant indicates distinct real roots, and a zero discriminant implies exactly one real root (or two equal real roots). This knowledge is invaluable in various mathematical contexts and has practical applications in fields such as physics, engineering, and computer science, where quadratic equations frequently arise. The ability to calculate and interpret the discriminant provides a powerful tool for analyzing and solving problems involving quadratic equations, allowing us to predict the nature of the solutions without explicitly solving for them. This understanding not only enhances our mathematical skills but also deepens our appreciation for the elegance and utility of quadratic equations in modeling and understanding the world around us. The discriminant, therefore, is more than just a formula; it is a gateway to unlocking the secrets of quadratic equations and their solutions.