Can someone please explain to me when why when the square term can be no less than 0, which means that it is the maximum? I understand that when the quadratic equation is negative, it means that the parabola is flipped upside down hence the y-value that is found is the maximum value, but I just did not understand the first way this solution was explained.

The wording of the solution indicates the following:

((x + 1/2)^2) cannot be less than 0: no square expression can ever be less than 0

- for a square number, the result is always either 0 itself or a positive number (try the three cases, squaring a negative number, 0, and a positive number, and demonstrate this property)

Because the equation is 5/4 - (x + 1/2)^2, notice a few things:

- We are subtracting a squared expression from 5/4. As detailed above, the value of the squared expression can either be 0 or a positive number. If you subtract a positive number from 5/4, the value would decrease. If you subtract 0 from 5/4, the value would remain 5/4. There is no case where the expression would give an output greater than 5/4, hence the wording that “5/4 is a maximum”

Hope this helps