What value needs to be added to the expression x^2 + x to transform it into a perfect square trinomial?

To determine the value that must be added to the expression x² + x to make it a perfect square trinomial, we can follow these steps:

1. Identify the form of a perfect square trinomial. A perfect square trinomial can be expressed as (a + b)² = a² + 2ab + b². In our case, we want to express x² + x in a similar manner.
2. Focus on the linear term. The linear term in our expression is x. To facilitate completing the square, we consider the coefficient of the x term, which is 1 in this case.
3. Calculate half of this coefficient. To find the necessary value, we take half of the coefficient of x, which is 1/2.
4. Square this value. Next, we square the result from the previous step: (1/2)² = 1/4.
5. Add this squared value to the expression. We now need to add 1/4 to the original expression, resulting in:

x² + x + 1/4

This can now be rewritten as a perfect square:

(x + 1/2)²

To summarize, the value that must be added to x² + x to make it a perfect square trinomial is 1/4.