Is The Discriminant Of G Positive Zero Or Negative

Is the Discriminant of g Positive, Zero, or Negative? A Comprehensive Guide

The discriminant, a powerful tool in algebra, provides crucial information about the nature of the roots (or solutions) of a quadratic equation. Understanding its implications is vital for solving various mathematical problems and interpreting real-world scenarios modeled by quadratic functions. This comprehensive guide will delve deep into the discriminant, exploring its calculation, interpretation, and applications, focusing on how to determine if it's positive, zero, or negative and what this signifies about the roots of a given quadratic equation represented by 'g'.

Understanding Quadratic Equations and Their Roots

Before diving into the discriminant, let's refresh our understanding of quadratic equations. A quadratic equation is an equation of the form:

g(x) = ax² + bx + c

where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The roots of this equation are the values of 'x' that make the equation equal to zero. These roots represent the x-intercepts of the parabola represented by the quadratic function g(x).

Introducing the Discriminant: The Key to Understanding Roots

The discriminant, denoted by Δ (Delta), is a part of the quadratic formula and is given by:

Δ = b² - 4ac

The discriminant provides valuable information about the nature of the roots without actually solving the quadratic equation. This makes it an incredibly efficient tool in various mathematical contexts.

Interpreting the Discriminant: Positive, Zero, or Negative?

The value of the discriminant directly dictates the type and number of roots the quadratic equation possesses:

1. Positive Discriminant (Δ > 0): Two Distinct Real Roots

If the discriminant is positive, the quadratic equation has two distinct real roots. Graphically, this means the parabola intersects the x-axis at two different points. These roots can be found using the quadratic formula:

x = (-b ± √Δ) / 2a

The "±" indicates that there are two separate solutions, one obtained by adding the square root of the discriminant and the other by subtracting it.

Example: Consider the equation g(x) = x² - 5x + 6. Here, a = 1, b = -5, and c = 6. The discriminant is:

Δ = (-5)² - 4(1)(6) = 25 - 24 = 1

Since Δ > 0, there are two distinct real roots. Using the quadratic formula, we find the roots to be x = 2 and x = 3.

2. Zero Discriminant (Δ = 0): One Repeated Real Root

If the discriminant is zero, the quadratic equation has one repeated real root (also called a double root). Graphically, this means the parabola touches the x-axis at exactly one point – the vertex of the parabola. The repeated root is given by:

x = -b / 2a

Example: Consider the equation g(x) = x² - 4x + 4. Here, a = 1, b = -4, and c = 4. The discriminant is:

Δ = (-4)² - 4(1)(4) = 16 - 16 = 0

Since Δ = 0, there is one repeated real root. Using the formula, we find the root to be x = 2.

3. Negative Discriminant (Δ < 0): Two Complex Conjugate Roots

If the discriminant is negative, the quadratic equation has two complex conjugate roots. These roots are complex numbers, meaning they involve the imaginary unit 'i' (where i² = -1). Complex conjugate roots always appear in pairs, in the form a + bi and a - bi, where 'a' and 'b' are real numbers. Graphically, this means the parabola does not intersect the x-axis.

Example: Consider the equation g(x) = x² + 2x + 5. Here, a = 1, b = 2, and c = 5. The discriminant is:

Δ = (2)² - 4(1)(5) = 4 - 20 = -16

Since Δ < 0, there are two complex conjugate roots. Using the quadratic formula, we find the roots to be x = -1 + 2i and x = -1 - 2i.

Applications of the Discriminant

The discriminant's power extends beyond simply finding the nature of roots. It finds applications in various fields:

• Geometry: Determining the nature of intersections between lines and conics (circles, parabolas, ellipses, hyperbolas). For instance, whether a line intersects a circle at two points, one point (tangent), or no points.

• Physics: Analyzing projectile motion, determining whether a projectile reaches a certain height or distance.

• Engineering: Designing structures and systems where stability and critical points are crucial. The discriminant can help determine the range of parameters for stable operation.

• Economics and Finance: Modeling various economic phenomena and making decisions based on the predicted outcome using quadratic models.

Beyond the Basics: Advanced Considerations

While the basic interpretation of the discriminant is straightforward, several advanced aspects warrant further exploration:

• Relationship to the Vertex: The x-coordinate of the vertex of a parabola is given by -b/2a. This is directly related to the repeated root when the discriminant is zero.

• Nature of Coefficients: The signs and magnitudes of the coefficients 'a', 'b', and 'c' influence the discriminant's value. Understanding these relationships can provide additional insights into the quadratic function's behavior.

• Numerical Stability: When dealing with very large or very small coefficients, numerical instability can affect the accuracy of discriminant calculations. Appropriate numerical techniques might be necessary for precise results.

Solving Problems Involving the Discriminant

Let's tackle some examples to solidify our understanding:

Problem 1: Determine the nature of the roots of the quadratic equation g(x) = 3x² - 7x + 2.

Solution: Here, a = 3, b = -7, and c = 2. The discriminant is:

Δ = (-7)² - 4(3)(2) = 49 - 24 = 25

Since Δ > 0, the quadratic equation has two distinct real roots.

Problem 2: For what values of 'k' does the equation g(x) = x² + kx + 9 have real roots?

Solution: For real roots, the discriminant must be greater than or equal to zero. Here, a = 1, b = k, and c = 9. Therefore:

k² - 4(1)(9) ≥ 0 k² ≥ 36 k ≥ 6 or k ≤ -6

Thus, the equation has real roots when k is greater than or equal to 6 or less than or equal to -6.

Problem 3: Find the value of 'm' such that the equation g(x) = mx² + 4x + 1 has exactly one root.

Solution: For exactly one root, the discriminant must be zero:

4² - 4(m)(1) = 0 16 - 4m = 0 m = 4

Therefore, the equation has exactly one root when m = 4.

Conclusion: Mastering the Discriminant

The discriminant is an invaluable tool for understanding the nature of the roots of a quadratic equation. Its ability to quickly and efficiently determine whether a quadratic equation possesses real, distinct, repeated, or complex roots makes it essential in various mathematical and real-world applications. By mastering the concepts and techniques discussed in this guide, you'll be well-equipped to analyze and solve a wide range of quadratic equations and related problems. Remember to always carefully calculate the discriminant and interpret its value correctly to gain valuable insights into the behavior of the quadratic function.