Complete the square f \( x^{2}+x+1 \)

To complete the square for the expression \( f(x) = x^2 + x + 1 \), you start by focusing on the \( x^2 + x \) part. Take half the coefficient of \( x \) (which is 1), square it (giving \( \frac{1}{4} \)), and then rewrite the equation: \[ f(x) = (x^2 + x + \frac{1}{4}) + 1 - \frac{1}{4} = \left(x + \frac{1}{2}\right)^2 + \frac{3}{4}. \] This shows that the expression can be rewritten as a perfect square plus a constant. Now, let's see how this completion of the square can be applied to graphing quadratic functions. By writing \( f(x) \) in its completed square form, \( f(x) = \left(x + \frac{1}{2}\right)^2 + \frac{3}{4} \), we can easily identify the vertex of the parabola, which is at the point \(-\frac{1}{2}, \frac{3}{4}\). This insight helps us sketch the graph and understand its minimum value—making it crucial for problem-solving in algebra and calculus!