Calculating Discriminant D Using The Formula B² - 4ac

Jul 8, 2025 by qnaftunila 54 views

Calculating Discriminant (d) Using the Quadratic Formula: A Step-by-Step Guide

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 the equation. The discriminant, often denoted as 'd', is a part of the quadratic formula that provides valuable insights into whether the quadratic equation has real and distinct roots, real and equal roots, or complex roots. In this comprehensive guide, we will delve into the discriminant, its significance, and how to calculate it using the formula d = b² - 4ac. We will also work through a specific example where we are given the values of a, b, and c, and tasked with calculating d. This guide aims to provide a clear understanding of the concept and equip you with the skills to apply the formula effectively.

Understanding the Discriminant

The discriminant (d), mathematically expressed as d = b² - 4ac, is a crucial component derived from the quadratic formula. It acts as a determinant, revealing the nature of the roots of a quadratic equation, which takes the standard form of ax² + bx + c = 0, where a, b, and c are constants. The value of the discriminant holds the key to understanding the characteristics of the solutions or roots of the quadratic equation. By calculating the discriminant, we can ascertain whether the quadratic equation has two distinct real roots, one repeated real root, or two complex roots. This knowledge is instrumental in various mathematical and scientific applications, as it allows us to predict the behavior of the quadratic equation without actually solving it.

The Significance of the Discriminant's Value

The significance of the discriminant lies in its ability to reveal the nature of the roots of a quadratic equation without the need to solve the equation completely. The value of d determines the type and number of solutions. This is crucial in many applications, including physics, engineering, and computer science, where understanding the nature of solutions is essential for modeling and predicting real-world phenomena. For instance, in physics, the discriminant can help determine the trajectory of a projectile, while in engineering, it can be used to analyze the stability of a system. The discriminant, therefore, is not just a theoretical concept but a practical tool with wide-ranging applications.

If d > 0: The quadratic equation has two distinct real roots. This means that there are two different real numbers that satisfy the equation. Graphically, this corresponds to the parabola intersecting the x-axis at two distinct points. These types of equations often represent scenarios where there are two possible solutions or outcomes.
If d = 0: The quadratic equation has one real root (a repeated root). This signifies that the equation has a single real number as a solution, which occurs twice. Graphically, the parabola touches the x-axis at exactly one point. This situation often arises when there is a critical point or a condition of tangency.
If d < 0: The quadratic equation has two complex roots. Complex roots involve imaginary numbers, indicating that there are no real number solutions. Graphically, the parabola does not intersect the x-axis. This outcome is common in scenarios where solutions involve oscillations or damped systems.

The Formula: d = b² - 4ac

The discriminant formula, d = b² - 4ac, is a straightforward yet powerful equation that allows us to compute the discriminant using the coefficients of the quadratic equation. Here, a represents the coefficient of the x² term, b is the coefficient of the x term, and c is the constant term. These coefficients are directly derived from the standard form of the quadratic equation, ax² + bx + c = 0. To calculate the discriminant, we simply substitute the given values of a, b, and c into the formula and perform the arithmetic operations. The result, the value of d, then provides the necessary information to determine the nature of the roots of the quadratic equation.

Breaking Down the Formula

To effectively use the discriminant formula, it's essential to understand each component and its role in determining the value of d. The formula d = b² - 4ac may seem simple, but each term contributes significantly to the final result.

b²: This term squares the coefficient of the x term. Squaring b ensures that the term is always positive, regardless of whether b is positive or negative. This term represents the central tendency or the balance of the quadratic equation. A larger b² value implies a greater influence of the linear term on the nature of the roots.
-4ac: This term multiplies the coefficient of the x² term (a) by the constant term (c) and then by -4. The product ac represents the interaction between the quadratic and constant terms. Multiplying by -4 scales this interaction and determines its contribution to the discriminant. The sign of ac (positive or negative) significantly impacts the discriminant's value and, consequently, the nature of the roots.

By carefully calculating and combining these components, we can accurately determine the value of the discriminant and infer the characteristics of the roots of the quadratic equation.

Step-by-Step Calculation with Given Values

Now, let's apply the discriminant formula to a specific example. We are given the following values: a = -9, b = -12, and c = 7. Our objective is to calculate d using the formula d = b² - 4ac. By following a step-by-step approach, we can ensure accuracy and clarity in our calculation. This example will demonstrate how to substitute the given values into the formula, perform the necessary arithmetic operations, and arrive at the final value of the discriminant.

Substituting the Values

The first step in calculating the discriminant is to substitute the given values of a, b, and c into the formula d = b² - 4ac. Given that a = -9, b = -12, and c = 7, we replace these variables in the formula:

d = (-12)² - 4(-9)(7)

This substitution sets up the equation for the subsequent arithmetic operations. It is crucial to ensure that the values are substituted correctly, paying close attention to the signs. A simple mistake in substitution can lead to an incorrect result, affecting the interpretation of the roots of the quadratic equation.

Performing the Arithmetic Operations

After substituting the values, the next step involves performing the arithmetic operations in the correct order, following the standard mathematical conventions (PEMDAS/BODMAS). The equation is:

d = (-12)² - 4(-9)(7)

First, we calculate the square of -12:

(-12)² = 144

Next, we multiply -4 by -9 and 7:

-4(-9)(7) = 36 * 7 = 252

Now, we substitute these values back into the equation:

d = 144 + 252

Finally, we add 144 and 252:

d = 396

Thus, the discriminant d is calculated to be 396. This value will then be used to determine the nature of the roots of the quadratic equation.

Final Result: d = 396

After performing the arithmetic operations, we arrive at the final result: d = 396. This value represents the discriminant for the quadratic equation with the given coefficients a = -9, b = -12, and c = 7. Now, we can interpret this result to understand the nature of the roots of the quadratic equation. Since d = 396 is greater than 0, the quadratic equation has two distinct real roots. This means there are two different real number solutions to the equation, and the parabola representing the equation will intersect the x-axis at two distinct points. This outcome is significant in various applications, providing crucial insights into the behavior of the system or phenomenon modeled by the quadratic equation.

Interpreting the Result

Having calculated the discriminant as d = 396, we can now interpret this value to determine the nature of the roots of the quadratic equation. The interpretation is based on the discriminant's value relative to zero. As we established earlier, the sign of the discriminant provides valuable information about the solutions of the quadratic equation. In this case, d = 396 is a positive number, which indicates a specific type of solution.

Two Distinct Real Roots

Since d = 396 is greater than 0, we conclude that the quadratic equation has two distinct real roots. This means that there are two different real numbers that satisfy the equation. Graphically, this corresponds to the parabola intersecting the x-axis at two distinct points. The existence of two real roots often implies that there are two possible solutions or outcomes in the context of the problem being modeled. For instance, in physics, this could represent two different points in time when a projectile reaches a certain height, or in engineering, it might indicate two different operating conditions for a system.

Conclusion

In this comprehensive guide, we have explored the concept of the discriminant, its formula, and its significance in determining the nature of the roots of a quadratic equation. We have demonstrated how to calculate the discriminant using the formula d = b² - 4ac and how to interpret the result. By working through a specific example where a = -9, b = -12, and c = 7, we calculated the discriminant to be 396 and concluded that the quadratic equation has two distinct real roots. This knowledge is invaluable in various fields, allowing us to predict and understand the behavior of quadratic equations in mathematical and real-world contexts. The discriminant, therefore, is a fundamental tool in algebra and a gateway to deeper insights into the solutions of quadratic equations.