Quadratic Applications

Most quadratic word problems ask one of two questions, and each has its own tool.

"What is the highest/lowest/best?" → find the vertex.
"When does it hit the ground / reach zero / break even?" → find the zeros.

A projectile is modelled by h(t) = −16t² + v₀t + h₀ in feet, or −4.9t² + v₀t + h₀ in metres. The negative leading coefficient is gravity, which is why the graph opens down.

Two answers often come out of the algebra when only one makes sense. Negative time, negative length and negative quantity are usually rejected. Say why you are rejecting it.

Applications reduce to reading a quadratic model's features. The vertex gives the optimum: the maximum for a < 0 (height, revenue) and the minimum for a > 0 (cost, distance). Its input is t = −b/(2a) and its output is the optimal value — and confusing the two is the most common error, since "when" wants the input and "how high" wants the output.

Zeros give boundary events: ground contact for a projectile, break-even points for profit P = R − C.

Domain restrictions from context are part of the answer. Time is typically restricted to [0, t_ground], and a length or quantity to positive values, so a negative root is discarded on physical grounds rather than mathematical ones.

Revenue models frequently arise from a linear demand relationship: if price p reduces quantity linearly, then R = p·q(p) is quadratic in p, and its vertex is the revenue-maximising price. Recognising that a product of two linear expressions is quadratic is what makes these problems tractable.