In all honesty, the only time I have provoked true enthusiasm - in contrast to genuine interest - is when I invented a science fiction story involving a dying planet, conversion of mass to energy (and vice versa), escape velocities etc. As I moved along on the math front, I incorporated new problems, using basic physics formulas and concepts, in my story. It was very time-consuming.

Among the many things I don't know how to do is to connect algebra to things kids find interesting. My story was a way to try to link math to science in a way that would work for teaching. It's hard for kids to use math to discover anything genuine, and even when it is possible, the students don't get enough practice to internalize any concept. I have thought about teaching the history of science and math together - many early experiments and observations are simple to repeat. I would like to think more about statistics/data analysis/algebra and teaching.

(I'm still interested in interesting quadratic problems.)

https://www.cut-the-knot.com/arithmetic/Means.shtml

The second could be modified to yield a quadratic equation. Say, a fellow jogs daily from home to the Central Park and back. On weekdays, he keeps constant speed, but on Saturdays and Sundays, he runs a little faster to the park, and slower by the same amount back home. Find that amount if the average speed was such-and-such. What average speeds are possible?

It's essential to consider the two problems simultaneously.