Question 857230
Maybe the wording has thrown your concept off?  Maybe you forgot that the question was about "range", and you momentarily mistook at as something else?


The wording, the quadratic equation x^2 -x +3 = k has real roots, is like f(x)=x^2-x+3, and then renaming f(x) as k.
NOW, what is the range for f(x)?  That is all the question asks.


Note that your function based on qualitative familiarity with this type of function, has a minimum and is concave upward.  The vertex is a MINIMUM; all values of the function are GREATER THAN OR EQUAL to the minimum value, which you found as 11/4.