## Understanding (x + 1/2)^2

The expression (x + 1/2)^2 represents the **square of the binomial (x + 1/2)**. In simpler terms, it means multiplying the binomial by itself.

### Expanding the Expression

To understand the expression better, we can expand it using the **FOIL method** (First, Outer, Inner, Last):

(x + 1/2)^2 = (x + 1/2)(x + 1/2)

**First:**x * x = x^2**Outer:**x * 1/2 = 1/2x**Inner:**1/2 * x = 1/2x**Last:**1/2 * 1/2 = 1/4

Combining the terms, we get:

(x + 1/2)^2 = **x^2 + x + 1/4**

### Applications of (x + 1/2)^2

This expression has applications in various areas of mathematics, including:

**Algebra:**Simplifying equations, solving quadratic equations, and finding the roots of polynomials.**Calculus:**Finding derivatives and integrals of functions.**Geometry:**Calculating areas and volumes of geometric shapes.

### Visual Representation

We can visualize (x + 1/2)^2 as a square with sides of length (x + 1/2). The area of this square is then represented by the expanded form:

**x^2:**Represents the area of a square with side length x.**x:**Represents the area of two rectangles with sides x and 1/2.**1/4:**Represents the area of a square with side length 1/2.

### Conclusion

Understanding the expansion and applications of (x + 1/2)^2 is crucial for solving various mathematical problems and gaining a deeper understanding of algebraic concepts. It's a fundamental expression that serves as a building block for more complex mathematical ideas.