Solve quadratic equations (factoring, completing the square, quadratic formula); interpret roots. | Algebra | ACT Mathematics

Use the quickest quadratic method and interpret the roots in context.

核心知识

Choose the quickest method the quadratic allows. Factor when the expression is friendly, complete the square when the form suggests it, and use the quadratic formula when the roots are not easy to see.

深入理解

Rule: A quadratic can have two real roots, one repeated root, or no real roots. On ACT, you are often deciding both how to solve it and what the roots mean.

In context, not every algebraic root is usable. A negative time, length, or count may solve the equation but fail the situation, so the final interpretation matters just as much as the solving method.

分步讲解

Set the quadratic equal to zero.
Choose the easiest method: factoring, completing the square, or the quadratic formula.
Check whether each root fits the context.

常见误解

Using a slower method when the quadratic factors cleanly.
Accepting a root that does not fit the situation, such as a negative time or count.
Forgetting that a quadratic can have two, one, or no real roots.

题目

示例解析

A ball's height is modeled by $ℎ = - 16 𝑡 2 + 48 𝑡 + 64$ , where $ℎ$ is in feet and $𝑡$ is in seconds. When does the ball hit the ground?

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