I’m guessing this should be -t²+(4/3)t+1/4.
Rewrite this: 1/4-(t²-(4/3)t)=1/4-(t²-(4/3)t+4/9-4/9)=
1/4-((t-(2/3))²-4/9)=1/4+4/9-(t-(2/3))²=25/36-(t-⅔)².
When 25/36-(t-⅔)²=0, the grasshopper reaches the ground, so
t-⅔=5/6 and t=⅔+5/6=3/2 seconds (1.5 seconds).
Another way to do this:
-t²+(4/3)t+1/4=0, -12t²+16t+3=0, 12t²-16t-3=0=(2t-3)(6t+1), so 2t-3=0, t=3/2.