For The Quadratic Equation $ax^2 + Bx + C = 0$ Where $a$ Is Not Equal To 0, If The Roots Are Real And Unequal, What Is True About The Value Of The Discriminant, $D$?

In mathematics, particularly in algebra, the discriminant plays a pivotal role in determining the nature of the roots of a quadratic equation. A quadratic equation, in its standard form, is expressed as ax² + bx + c = 0, where a, b, and c are coefficients, and a ≠ 0. The discriminant, often denoted by D, is a formula derived from these coefficients that provides valuable information about the roots of the equation without actually solving for them. The discriminant is calculated using the formula: D = b² - 4ac. The value of D reveals whether the roots are real or complex, and if real, whether they are distinct or repeated. Understanding the discriminant is crucial for solving quadratic equations and for various applications in mathematics, physics, engineering, and other fields. The following sections will delve into the specific cases of the discriminant and their corresponding root characteristics.

The discriminant, D = b² - 4ac, is a powerful tool in determining the nature of the roots of a quadratic equation. When the discriminant is greater than zero (D > 0), it indicates that the quadratic equation has two distinct real roots. This means that there are two different values of x that satisfy the equation. Geometrically, this corresponds to the parabola represented by the quadratic equation intersecting the x-axis at two distinct points. The larger the value of D, the greater the separation between the two roots. In practical terms, this scenario often arises in physical problems where two distinct solutions are possible, such as in projectile motion or electrical circuits. For instance, if we consider the equation x² - 5x + 6 = 0, the discriminant D is calculated as (-5)² - 4(1)(6) = 25 - 24 = 1, which is greater than zero. Thus, this equation has two distinct real roots, which can be found by factoring the quadratic equation into (x - 2)(x - 3) = 0, yielding the roots x = 2 and x = 3. These roots represent the points where the parabola y = x² - 5x + 6 intersects the x-axis. The condition D > 0 is not only a mathematical criterion but also a gateway to understanding the physical implications and interpretations of quadratic equations in various scientific and engineering contexts. Therefore, recognizing this condition is fundamental in solving problems where distinct real solutions are expected and applicable.

When the discriminant equals zero (D = 0), the quadratic equation has exactly one real root, which is a repeated root. This occurs because the quadratic formula, which gives the roots as x = (-b ± √D) / (2a), simplifies to x = -b / (2a) when D is zero. The repeated root represents a single point where the parabola touches the x-axis, rather than crossing it at two distinct points. This situation is significant in various applications. For example, in physics, it might represent a critical damping condition in a damped harmonic oscillator, where the system returns to equilibrium as quickly as possible without oscillating. In mathematics, a repeated root implies that the quadratic expression is a perfect square, such as (x - r)² = 0, where r is the repeated root. Consider the equation x² - 4x + 4 = 0. Here, the discriminant D is calculated as (-4)² - 4(1)(4) = 16 - 16 = 0. This indicates that the equation has a single repeated real root. Indeed, the equation can be factored as (x - 2)² = 0, giving the repeated root x = 2. Graphically, the parabola y = x² - 4x + 4 touches the x-axis at the single point x = 2. Understanding the D = 0 condition is essential for identifying scenarios where there is a unique solution, especially in contexts where multiple distinct solutions would not make physical or practical sense. The concept of a repeated root is a cornerstone in the broader theory of polynomial equations and their solutions.

Conversely, if the discriminant is less than zero (D < 0), the quadratic equation has no real roots. In this case, the roots are complex numbers, which involve the imaginary unit i, where i² = -1. Complex roots occur in conjugate pairs, meaning that if a + bi is a root, then a - bi is also a root, where a and b are real numbers. The absence of real roots implies that the parabola represented by the quadratic equation does not intersect the x-axis. This situation is commonly encountered in mathematical models of physical systems that exhibit oscillatory behavior without damping, such as an ideal harmonic oscillator or in the analysis of alternating current circuits with reactive components. For example, consider the quadratic equation x² + 2x + 5 = 0. The discriminant D is calculated as (2)² - 4(1)(5) = 4 - 20 = -16, which is less than zero. This indicates that the equation has complex roots. Using the quadratic formula, the roots can be found as x = (-2 ± √(-16)) / 2 = -1 ± 2i. These complex roots, -1 + 2i and -1 - 2i, do not correspond to any real number solutions on the x-axis. Graphically, the parabola y = x² + 2x + 5 lies entirely above the x-axis, demonstrating that there are no real solutions. Recognizing the D < 0 condition is crucial for identifying scenarios where real-number solutions do not exist, and understanding complex numbers becomes necessary to fully solve and interpret the mathematical problem. This condition often points to underlying physical constraints or assumptions in the model that prevent real-world solutions.

In the context of quadratic equations, the relationship between the discriminant and the nature of the roots is fundamental. The question posed asks about the condition for the roots of the quadratic equation ax² + bx + c = 0, where a ≠ 0, to be real and unequal. This directly corresponds to the case where the discriminant, D, is greater than zero (D > 0). As previously explained, when D > 0, the quadratic equation has two distinct real roots. This is because the square root of a positive number in the quadratic formula yields two different real values, leading to two distinct solutions for x. Therefore, among the options provided: (a) D < 0, (b) D > 0, (c) D = 0, and (d) D ≤ 0, the correct answer is (b) D > 0. The other options are incorrect because D < 0 implies complex roots, D = 0 implies a repeated real root, and D ≤ 0 implies either a repeated real root or complex roots. The condition D > 0 is the specific criterion that ensures the roots are both real and unequal. This understanding is critical not only for solving quadratic equations but also for analyzing and interpreting mathematical models in various scientific and engineering applications. Recognizing this relationship allows one to quickly determine the nature of the roots without having to solve the entire equation, thereby saving time and providing valuable insights into the problem at hand.

In conclusion, the discriminant is a powerful tool for understanding the nature of the roots of a quadratic equation. When the roots of the quadratic equation ax² + bx + c = 0 are real and unequal, it directly implies that the discriminant, D, must be greater than zero (D > 0). This condition ensures that the quadratic equation has two distinct real solutions. The cases where D < 0 result in complex roots, and D = 0 results in a single, repeated real root. Therefore, for real and unequal roots, the condition D > 0 is both necessary and sufficient. This understanding is fundamental in algebra and has wide-ranging applications in various fields, including physics, engineering, and computer science. By simply calculating the discriminant, one can quickly determine the nature of the roots without needing to fully solve the quadratic equation, making it an invaluable concept in mathematical analysis and problem-solving.