Solve y=x²+2x+2: Finding Where Function Values Are Positive

To find for which values of $x$ the function $y = x^2 + 2x + 2$ is positive, we'll begin by analyzing the quadratic expression.

First, let's complete the square for the quadratic function:

To complete the square, take the coefficient of the $x$ term (which is 2), divide it by 2 to get 1, and then square it to obtain 1. Add and subtract this value inside the expression:

This simplifies to:

Now, this function $y = (x+1)^2 + 1$ shows a parabola in the vertex form $(x-h)^2 + k$, where the vertex is at $(-1, 1)$ and opens upwards, as the coefficient of the squared term is positive.

This indicates that the minimum point, $\text{(vertex)}$, is at $y = 1$, which is above the x-axis. As such, the function $y = (x+1)^2 + 1$ will be positive for all $x$ since the entire curve lies above the x-axis and the value of $y$ is always greater than zero.

Therefore, the function $y = x^2 + 2x + 2$ is positive for all real values of $x$.

Hence, the solution is that the function is positive for all values of $x$.