How to solve the following quadratic word problem given a quadratic equation?

Question

The height of a ball $(h)$ feet, after $s$ seconds is modeled by the equation: $$h=-16t^2+40t-6$$ How many seconds does it take for the ball ($t$) reach its maximum height? First thing I did was turn the negatives into positive. And also factored it $$h=2(8t^2 - 20t + 3)$$ But then I couldn't find the factors that would give me $20$? How would I solve this problem?

Answer 1

Don't turn the negatives into positives, but do complete the square.

$$\begin{align} h&=-16t^2+40t-6 \\ &= -16\left(t^2-\frac 52t\right)-6 \\ &= -16\left(t^2-\frac 52t+\frac{25}{16}\right)-6+25 \\ &= -16\left(t-\frac 54\right)^2+19 \\ \end{align}$$

We see the maximum happens at $t=\frac 54$ and the maximum height is $19$.

Answer 2

Given the exact thing you must solve, the time for maximum height, the simplest, fastest solution is to find the vertex, using $x=\frac{-b}{2a}$, this results in $\frac{-40}{-32} = \frac{5}{4}$ seconds