Solve any quadratic equation with the quadratic formula and use the discriminant to predict the number and type of solutions.
Complete the square on the GENERAL equation ax^2+bx+c=0 (instead of a specific one) and you get a formula that solves every quadratic equation that has ever existed or ever will: x = (-b +- sqrt(b^2-4ac)) / (2a). Memorize it, but also trust it -- it never fails, even when factoring is impossible.
The expression under the root, b^2-4ac, is called the discriminant, and it tells you what kind of answer to expect BEFORE you finish solving. Positive discriminant: two different real solutions. Zero discriminant: exactly one real solution (a repeated root -- the parabola just touches the x-axis). Negative discriminant: no real solutions, but two complex solutions, which we tackle next week.
Mastery looks like: They apply the quadratic formula correctly to any quadratic and use the discriminant to predict the number and type of solutions before or without fully solving.
Common stumbles: Forgetting parentheses around a negative b or c when substituting (losing a sign); computing the discriminant but then ignoring what it predicts.