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

Expansion: The lefthand 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

Standard Form: Therefore, the expanded form of the equation is: x² + 8x + 15 = y

Parabola: This equation represents a parabola, which is a symmetrical Ushaped 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

Vertex: The vertex is the lowest point on the parabola. Its xcoordinate 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 xcoordinate of the vertex is 8 / 2(1) = 4. To find the ycoordinate, we substitute x = 4 into the equation: (4)² + 8(4) + 15 = 1. The vertex is (4, 1).

XIntercepts: These are the points where the parabola intersects the xaxis (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 xintercepts are x = 3 and x = 5.

YIntercept: This is the point where the parabola intersects the yaxis (where x = 0). Substituting x = 0 into the equation, we get y = 0² + 8(0) + 15 = 15. The yintercept 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, xintercepts, and yintercept. By understanding the equation and its features, we can analyze the relationship between variables and utilize it in various applications.