(m-2)x2-(5+m)x+16

4 min read Jun 16, 2024
(m-2)x2-(5+m)x+16

Exploring the Quadratic Expression: (m-2)x² - (5+m)x + 16

This article will delve into the quadratic expression (m-2)x² - (5+m)x + 16 and discuss its properties, potential applications, and how to analyze it.

Understanding the Expression

The given expression is a quadratic equation in the variable x, with m as a parameter. This means that the value of m will influence the behaviour of the quadratic expression. We can observe the following key features:

  • Leading Coefficient: (m-2) determines the shape of the parabola representing the expression. If (m-2) is positive, the parabola opens upwards, and if it's negative, it opens downwards.
  • Linear Coefficient: -(5+m) affects the position of the vertex and the axis of symmetry.
  • Constant Term: 16 is the y-intercept of the parabola.

Analyzing the Expression

To gain a deeper understanding of this expression, we can investigate several aspects:

  • Roots or Solutions: Finding the roots or solutions (values of x for which the expression equals zero) involves solving the quadratic equation. We can use the quadratic formula for this:

    x = [-b ± √(b² - 4ac)] / 2a

    where a = (m-2), b = -(5+m), and c = 16.

  • Vertex: The vertex of the parabola is the point where the expression reaches its maximum or minimum value. Its coordinates can be determined using:

    x-coordinate of vertex = -b / 2a

    y-coordinate of vertex = f(-b / 2a)

    where f(x) represents the quadratic expression.

  • Discriminant: The discriminant (b² - 4ac) provides insights into the nature of the roots:

    • If the discriminant is positive, there are two distinct real roots.
    • If the discriminant is zero, there is one repeated real root.
    • If the discriminant is negative, there are two complex roots.

Applications

This type of quadratic expression can find applications in various fields, such as:

  • Physics: Modelling projectile motion or oscillations.
  • Engineering: Designing curves for roads or bridges.
  • Economics: Analyzing profit functions or supply and demand curves.

Conclusion

The quadratic expression (m-2)x² - (5+m)x + 16 represents a versatile mathematical concept with applications in various disciplines. Understanding its properties and how to analyze it empowers us to solve problems and gain insights into real-world phenomena. By exploring its roots, vertex, and discriminant, we can unlock a deeper understanding of its behavior and potential applications.

Related Post


Featured Posts