(x%2B3)%3Dy.jpeg "(x+5)(x+3)=y")
Exploring the Equation: (x+5)(x+3) = y
This equation represents a quadratic relationship between the variables x and y. Let's break down its meaning and explore its key features.
Understanding the Equation
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Expansion: The left-hand side of the equation involves multiplying two binomials: (x+5) and (x+3). Using the FOIL method (First, Outer, Inner, Last), we expand it as follows:
- First: x * x = x²
- Outer: x * 3 = 3x
- Inner: 5 * x = 5x
- Last: 5 * 3 = 15
- Combining terms: x² + 3x + 5x + 15 = x² + 8x + 15
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Standard Form: Therefore, the expanded form of the equation is: x² + 8x + 15 = y
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Parabola: This equation represents a parabola, which is a symmetrical U-shaped curve. The coefficient of the x² term (1 in this case) determines whether the parabola opens upwards (positive) or downwards (negative). In our equation, the parabola opens upwards.
Key Features of the Equation
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Vertex: The vertex is the lowest point on the parabola. Its x-coordinate can be found using the formula: x = -b / 2a, where a and b are the coefficients of the quadratic equation (in this case, a = 1 and b = 8). Therefore, the x-coordinate of the vertex is -8 / 2(1) = -4. To find the y-coordinate, we substitute x = -4 into the equation: (-4)² + 8(-4) + 15 = -1. The vertex is (-4, -1).
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X-Intercepts: These are the points where the parabola intersects the x-axis (where y = 0). To find them, we set y = 0 and solve the quadratic equation: x² + 8x + 15 = 0. Factoring the equation, we get (x+3)(x+5) = 0. Therefore, the x-intercepts are x = -3 and x = -5.
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Y-Intercept: This is the point where the parabola intersects the y-axis (where x = 0). Substituting x = 0 into the equation, we get y = 0² + 8(0) + 15 = 15. The y-intercept is (0, 15).
Applications
This equation can be applied in various fields, such as:
- Physics: Describing the trajectory of projectiles.
- Engineering: Modeling the shape of certain structures like bridges and arches.
- Economics: Analyzing market trends and forecasting demand.
Conclusion
The equation (x+5)(x+3) = y represents a simple yet powerful quadratic relationship, describing a parabola with a distinct vertex, x-intercepts, and y-intercept. By understanding the equation and its features, we can analyze the relationship between variables and utilize it in various applications.