Discriminant Calculation For Quadratic Equation X^2 + 6x - 55 = 0
The quadratic formula stands as a cornerstone in algebra, a universal tool for solving quadratic equations of the form ax² + bx + c = 0. This formula, derived from the method of completing the square, provides the solutions, also known as roots or zeros, of any quadratic equation. The formula is expressed as:
Within this elegant equation lies a critical component known as the discriminant, represented by the expression b² - 4ac. The discriminant, nestled under the radical sign, holds the key to understanding the nature of the quadratic equation's solutions. It dictates whether the solutions are real or complex, distinct or repeated. When simplifying the quadratic formula to solve a specific equation, the value of the discriminant is the first piece of information we uncover, and in the case of the equation x² + 6x - 55 = 0, finding this value is our primary focus.
To unravel the mystery of the discriminant in the equation x² + 6x - 55 = 0, we must first identify the coefficients a, b, and c. In this particular quadratic equation, the coefficient of the x² term, a, is 1; the coefficient of the x term, b, is 6; and the constant term, c, is -55. With these values in hand, we can now embark on the journey of substituting them into the discriminant formula, b² - 4ac. By carefully performing the arithmetic operations, we will arrive at the numerical value of the discriminant, which will reveal the characteristics of the solutions to our equation. This process of substitution and simplification is not merely a mechanical exercise; it's a crucial step in gaining insight into the mathematical structure of the quadratic equation and the nature of its roots. The discriminant acts as a beacon, guiding us towards a deeper understanding of the equation's behavior and the properties of its solutions. By meticulously calculating the discriminant, we are laying the foundation for a comprehensive analysis of the quadratic equation x² + 6x - 55 = 0.
Calculating the Discriminant for x² + 6x - 55 = 0
The heart of our task lies in calculating the value of the discriminant, b² - 4ac, for the quadratic equation x² + 6x - 55 = 0. As we previously identified, a = 1, b = 6, and c = -55. Now, we embark on the process of substituting these values into the discriminant formula. Replacing b with 6, a with 1, and c with -55, we get:
Discriminant = 6² - 4(1)(-55)
This substitution sets the stage for the arithmetic dance that will reveal the discriminant's numerical value. We begin by evaluating the exponent, 6², which equals 36. Then, we address the multiplication: 4 multiplied by 1 and then by -55. This yields -220. However, the negative sign in front of the 4 in the formula transforms this -220 into +220. Now, our equation looks like this:
Discriminant = 36 + 220
The final step is a simple addition. Adding 36 and 220, we arrive at the discriminant's value:
Discriminant = 256
This numerical result, 256, is the number that resides under the radical sign when we simplify the quadratic formula for the given equation. It's not just a number; it's a piece of vital information about the nature of the solutions. The positive value of the discriminant tells us that the quadratic equation has two distinct real roots. If the discriminant were zero, the equation would have one real root (a repeated root), and if it were negative, the equation would have two complex roots. Therefore, the discriminant serves as a crucial indicator, providing insights into the characteristics of the solutions without actually solving the entire quadratic formula. In this case, the discriminant's value of 256 confirms that we will find two different real number solutions for x when we fully solve the equation x² + 6x - 55 = 0 using the quadratic formula.
Interpreting the Discriminant: Unveiling the Nature of Solutions
The discriminant, the value under the radical in the quadratic formula, acts as a powerful indicator of the nature of the solutions to a quadratic equation. Its sign, whether positive, negative, or zero, dictates the type and number of roots the equation possesses. Understanding this relationship is crucial for gaining a comprehensive understanding of quadratic equations. In our case, we calculated the discriminant for the equation x² + 6x - 55 = 0 and found it to be 256, a positive value. This positive discriminant immediately tells us that the equation has two distinct real roots. This means that there are two different real numbers that, when substituted for x in the equation, will make the equation true.
To delve deeper into the significance of a positive discriminant, let's consider the broader context of discriminant interpretation. When the discriminant (b² - 4ac) is greater than zero, as in our example, the quadratic equation has two distinct real roots. Graphically, this corresponds to the parabola representing the quadratic equation intersecting the x-axis at two distinct points. These points of intersection are the real roots of the equation. On the other hand, if the discriminant is equal to zero, the quadratic equation has exactly one real root, often referred to as a repeated root. In this scenario, the parabola touches the x-axis at only one point, indicating that the single root is a double root. The final possibility is when the discriminant is less than zero, meaning it's a negative value. In this case, the quadratic equation has no real roots; instead, it has two complex roots. Graphically, this means the parabola does not intersect the x-axis at all. The roots are complex numbers involving the imaginary unit 'i', where i² = -1. These complex roots always come in conjugate pairs.
Therefore, by simply calculating the discriminant, we can predict the nature of the solutions without going through the entire process of solving the quadratic formula. In the case of x² + 6x - 55 = 0, the positive discriminant of 256 assures us that we will find two different real number solutions for x. This knowledge allows us to approach the full solution with a clear understanding of what to expect, saving time and effort in the process. The discriminant, in essence, is a shortcut to understanding the fundamental characteristics of a quadratic equation's solutions.
Conclusion: The Power of the Discriminant
In conclusion, when simplifying the quadratic formula to solve the equation x² + 6x - 55 = 0, the number under the radical sign, the discriminant, is 256. This seemingly simple calculation unlocks a wealth of information about the equation's solutions. The discriminant, represented by b² - 4ac, serves as a crucial indicator of the nature and number of roots a quadratic equation possesses. By identifying the coefficients a, b, and c in the equation and substituting them into the discriminant formula, we can quickly determine whether the equation has two distinct real roots, one repeated real root, or two complex roots. This knowledge is invaluable as it allows us to anticipate the type of solutions we will obtain before embarking on the full solution process.
For the equation x² + 6x - 55 = 0, the positive discriminant of 256 unequivocally tells us that the equation has two distinct real roots. This means that there are two different real number values for x that will satisfy the equation. This information can guide our subsequent steps in solving the equation, helping us to choose the most efficient method and interpret the results accurately. The discriminant, therefore, is not merely a number; it's a powerful tool that provides a shortcut to understanding the fundamental characteristics of a quadratic equation's solutions.
Understanding the discriminant is a fundamental concept in algebra, particularly when dealing with quadratic equations. It allows us to analyze and classify quadratic equations based on the nature of their solutions. This understanding is not only crucial for solving mathematical problems but also for applying quadratic equations in various real-world scenarios, such as physics, engineering, and economics. By mastering the concept of the discriminant, we gain a deeper appreciation for the elegance and power of the quadratic formula and its applications. The ability to quickly determine the nature of solutions using the discriminant enhances our problem-solving skills and provides a solid foundation for more advanced mathematical concepts. Therefore, the discriminant is an indispensable tool in the algebraic toolbox, empowering us to effectively analyze and solve quadratic equations in a wide range of contexts.
