x2-(5%2Bm)x%2B16.jpeg "(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:
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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.
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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.
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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.