x%5E2%2B(2%2F9)x-8.jpeg "(1/81)x^2+(2/9)x-8")
Analyzing the Quadratic Expression: (1/81)x^2 + (2/9)x - 8
This article will explore the quadratic expression (1/81)x^2 + (2/9)x - 8, analyzing its key features and discussing methods to work with it.
Understanding the Expression
The expression (1/81)x^2 + (2/9)x - 8 is a quadratic expression because the highest power of the variable 'x' is 2. It is in the standard form: ax^2 + bx + c, where:
- a = 1/81 (the coefficient of the x^2 term)
- b = 2/9 (the coefficient of the x term)
- c = -8 (the constant term)
Key Features
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Leading Coefficient: The leading coefficient (a = 1/81) is positive, indicating that the parabola representing this expression opens upwards.
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Constant Term: The constant term (c = -8) represents the y-intercept, meaning the graph of this expression will cross the y-axis at the point (0, -8).
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Axis of Symmetry: The axis of symmetry is a vertical line that divides the parabola in half. Its equation is x = -b / 2a. Substituting the values for 'a' and 'b' into this equation will give us the x-coordinate of the vertex of the parabola.
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Vertex: The vertex is the minimum point of the parabola. Its x-coordinate is found using the axis of symmetry, and its y-coordinate can be obtained by substituting this x-value back into the original expression.
Solving for Roots
The roots of the quadratic expression are the x-values where the expression equals zero. This can be found using the quadratic formula:
x = [-b ± √(b^2 - 4ac)] / 2a
By substituting the values for 'a', 'b', and 'c', you can calculate the roots of the expression.
Applications
Quadratic expressions are widely used in various fields, including:
- Physics: To model the trajectory of projectiles.
- Engineering: To design bridges and structures.
- Finance: To calculate interest rates and returns on investments.
Conclusion
By understanding the key features and applying appropriate methods, we can analyze and manipulate the quadratic expression (1/81)x^2 + (2/9)x - 8. This knowledge is valuable for solving problems in various disciplines and gaining insights into the behavior of quadratic functions.