## Solving the Equation: (x + 1/2)^2 - (x + 1/2)(x + 6) = 8

This article will guide you through the steps to solve the given quadratic equation. We will use algebraic manipulation and the quadratic formula to find the solutions.

### Step 1: Expanding the Equation

First, we need to expand the equation by multiplying out the brackets:

**(x + 1/2)^2:**This expands to x² + x + 1/4**(x + 1/2)(x + 6):**This expands to x² + 6.5x + 3

Substituting these back into the original equation, we get:

x² + x + 1/4 - (x² + 6.5x + 3) = 8

### Step 2: Simplifying the Equation

Now, we simplify the equation by combining like terms:

x² + x + 1/4 - x² - 6.5x - 3 = 8

-5.5x - 11/4 = 8

### Step 3: Isolating the x Term

To isolate the x term, we move the constant term to the right side of the equation:

-5.5x = 8 + 11/4

-5.5x = 43/4

### Step 4: Solving for x

Finally, we solve for x by dividing both sides of the equation by -5.5:

x = (43/4) / (-5.5)

**x = -1.95** (approximately)

### Conclusion

Therefore, the solution to the equation (x + 1/2)^2 - (x + 1/2)(x + 6) = 8 is **x = -1.95**. You can verify this solution by substituting it back into the original equation.