Solve by factoring, completing the square, or the quadratic formula — and know which method is fastest for the equation in front of you. Every question comes with a written walkthrough of the method used and why it fit.
A quadratic almost always has two roots; dropping the minus branch loses half the answer.
Factoring only works when one side of the equation is zero — factor the wrong side and the zero-product property doesn't apply.
A negative length or negative time is mathematically valid but physically meaningless — the context, not the algebra, tells you to discard it.
Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.
Solve by factoring: x² − x − 6 = 0
A — x = 3 or x = −2. Factor: two numbers that multiply to −6 and add to −1 are −3 and 2, giving (x − 3)(x + 2) = 0, so x = 3 or x = −2. x = −3 or x = 2 (B) swaps the signs of both roots — the result of a plausible-looking but different factoring that doesn't actually multiply out to the original trinomial.
Solve using the quadratic formula: 2x² + 3x − 2 = 0
A — x = 0.5 or x = −2. With a = 2, b = 3, c = −2, the discriminant is 3² − 4(2)(−2) = 9 + 16 = 25, and √25 = 5. The formula gives x = (−3 ± 5) / 4, so x = 2/4 = 0.5 or x = −8/4 = −2. x = −0.5 or x = 2 (B) comes from flipping the sign on the ±5 branch assignment — pairing −3 with the wrong sign of the root on each branch.
A ball is launched upward with height h(t) = −16t² + 64t + 5 (feet, seconds). At what time, to two decimals, does it hit the ground?
A — 4.08 s. Set h(t) = 0: −16t² + 64t + 5 = 0. The quadratic formula gives two roots, one negative and one positive; only the positive one is physically meaningful since time can't be negative. That root is t ≈ 4.08 seconds. 4.00 s (B) is what you get by ignoring the "+5" initial height entirely and solving the simpler −16t² + 64t = 0 — close, but it drops real information from the problem.
Quadratics reward knowing which method to reach for — factoring when it's fast, the formula when it isn't — practice builds that judgment as much as the mechanics.
For every equation, spend five seconds checking whether it factors cleanly before defaulting to the quadratic formula. Untimed practice is where that judgment call gets fast.
Move to timed sessions once method selection is quick. Quadratics run longer than linear equations — budget the extra time.
Combine quadratic equations with quadratic functions in a mock — the algebra here is the same skill tested from a graphing angle there.
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