Let \(a,b,\) and \(t\) be real numbers such that \(a + b = t.\) Find, in terms of \(t,\) the minimum value of \(a^2 + b^2.\)

\(b=t-a\\ a^2+b^2 = a^2 + (t-a)^2 = \\ 2a^2 - 2at + t^2 = \\ 2\left(a-\dfrac t 2 \right)^2 + \dfrac{t^2}{2}\\ \text{This clearly has a minimum value of }\dfrac{t^2}{2} \text{ at }a = \dfrac t 2\)