Strategies to deepen intuition for quadratics beyond the quadratic formula.

[Robert_G]

I'm tutoring a high school student in algebra, and we've hit a wall with quadratic equations, specifically with her understanding of why the quadratic formula works and how it relates to factoring and completing the square. She can mechanically apply the formula, but if she makes a small arithmetic error, she has no conceptual backup to check her work or even estimate what the roots should be. For other educators or tutors, what are your most effective methods for building that deeper intuition about quadratics, perhaps using visual models or real-world applications, to move students beyond rote memorization? I'm looking for activities or explanations that connect the formula to the graph and the factored form in a meaningful way.