In order for Jasmine to have an average of 45 for all 20 class quizzes, she needs to have a total score of 45 * 20 = 900 points, out of a total 50 * 20 = 1000 points at stake.
An average of 35 on the first five quizzes means that she has earned a total of 35 * 5 = 175 points on those quizzes. These count toward the required 900, so this means that she needs to earn a total of 900 - 175 = 725 points on the next 15 quizzes. (The total points at stake for those 15 quizzes is 15 * 50 = 750.)
To get the lowest score on the sixth quiz and still end up having earned a total of 725 points on the last 15 quizzes (or 900 points on all 20 quizzes), she needs to earn a perfect 50 on all quizzes from the seventh one onward, which is 14 quizzes. Doing so will earn 14 * 50 = 700 points, leaving us 25 points short of the minimum required 725. This means that the lowest possible score that can be earned on the sixth quiz is 725 - 700 = 25 points.