What Is The Discriminant Of The Quadratic Equation 3-4x-6x2

What is the Discriminant of the Quadratic Equation 3 - 4x - 6x²? A Deep Dive

The discriminant of a quadratic equation is a powerful tool that reveals crucial information about the nature of its roots (solutions). Understanding the discriminant allows us to determine whether a quadratic equation has real or complex roots, and whether those roots are distinct or repeated. This article will delve into the concept of the discriminant, focusing specifically on the quadratic equation 3 - 4x - 6x², illustrating its calculation and interpreting its meaning.

Understanding Quadratic Equations and Their Roots

A quadratic equation is a polynomial equation of the second degree, generally expressed in the standard form:

ax² + bx + c = 0

where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The roots of the equation are the values of 'x' that satisfy the equation. These roots can be found using various methods, including factoring, completing the square, or the quadratic formula.

Introducing the Discriminant: The Key to Understanding the Roots

The discriminant (often denoted as Δ or D) is a part of the quadratic formula that sits inside the square root. It's defined as:

Δ = b² - 4ac

The discriminant's value directly impacts the nature of the quadratic equation's roots:

• Δ > 0 (Positive Discriminant): The quadratic equation has two distinct real roots. This means there are two different values of 'x' that satisfy the equation.

• Δ = 0 (Zero Discriminant): The quadratic equation has exactly one real root (a repeated root). This means the quadratic equation represents a perfect square.

• Δ < 0 (Negative Discriminant): The quadratic equation has two distinct complex roots (conjugate pairs). These roots involve the imaginary unit 'i' (√-1).

Applying the Discriminant to 3 - 4x - 6x²

Let's apply our knowledge to the specific quadratic equation: 3 - 4x - 6x². First, we need to rewrite the equation in standard form:

-6x² - 4x + 3 = 0

Now we can identify the coefficients:

• a = -6
• b = -4
• c = 3

Next, we substitute these values into the discriminant formula:

Δ = b² - 4ac = (-4)² - 4(-6)(3) = 16 + 72 = 88

Since Δ = 88, which is greater than 0, we conclude that the quadratic equation 3 - 4x - 6x² has two distinct real roots.

Solving for the Roots: Illustrating the Discriminant's Impact

Now that we know the nature of the roots, let's find their actual values using the quadratic formula:

x = (-b ± √Δ) / 2a

Substituting the values of a, b, and Δ:

x = (4 ± √88) / (2 * -6) = (4 ± 2√22) / -12 = (-2 ± √22) / 6

Therefore, the two distinct real roots are:

• x₁ = (-2 + √22) / 6
• x₂ = (-2 - √22) / 6

Visualizing the Roots: A Graphical Interpretation

The graph of a quadratic equation is a parabola. The x-intercepts of the parabola represent the roots of the equation. Since our discriminant is positive, the parabola intersects the x-axis at two distinct points, confirming our findings. A negative discriminant would result in a parabola that doesn't intersect the x-axis at all, while a zero discriminant would indicate the parabola's vertex touches the x-axis at only one point.

The Discriminant's Significance in Different Contexts

The discriminant's applications extend beyond simply determining the nature of roots. It plays a crucial role in various mathematical and scientific fields:

• Physics: In projectile motion, the discriminant can determine whether a projectile will ever reach a certain height.

• Engineering: The discriminant is used in stability analysis of systems, indicating whether a system is stable, unstable, or marginally stable.

• Computer Graphics: The discriminant is utilized in algorithms for ray-tracing and collision detection.

• Economics: Quadratic equations and their discriminants appear in economic models, particularly in optimization problems.

Advanced Applications and Extensions

The concept of the discriminant can be extended to higher-degree polynomial equations, though the formulas become significantly more complex. The discriminant of a cubic equation, for instance, involves a more intricate expression of its coefficients. However, the underlying principle remains the same: the discriminant provides critical information about the nature and number of the equation's roots. Numerical methods are often employed to find the roots of higher-degree polynomials efficiently, with the discriminant still providing valuable preliminary insights into the expected properties of those roots.

Conclusion: The Discriminant's Enduring Power

The discriminant is a fundamental concept in algebra with far-reaching implications. Its simplicity belies its power, allowing us to quickly and efficiently assess the nature of a quadratic equation's roots without needing to fully solve the equation. This understanding of the discriminant is essential for anyone working with quadratic equations, regardless of their field of study or profession. Its ability to provide immediate insight into the properties of solutions makes it a cornerstone of algebraic analysis and a valuable tool across numerous disciplines. From simple problem-solving to complex mathematical modeling, the discriminant continues to hold significance as a fundamental tool for understanding and interpreting quadratic equations. The specific example of 3 - 4x - 6x² perfectly demonstrates the straightforward yet powerful application of this important concept.