A ball is thrown upward from the top of a 20-meter-high building. Its height h h in meters after t t seconds is given by: h ( t ) = − 5 t 2 + 15 t + 20 h(t)=−5t 2 + $15t+20$

Step 1: Set height to zero

When the ball hits the ground, its height h(t) will be 0 meters. We set the given height equation equal to 0 to find the time $t$ when this occurs.

Step 2: Formulate the quadratic equation

Substitute $h(t) = 0$ into the given equation to form a quadratic equation in terms of $t$. This equation will allow us to solve for the time $t$.

Step 3: Simplify the quadratic equation

To simplify the equation, we can divide the entire equation by $-5$. This makes the coefficients smaller and easier to work with, without changing the roots of the equation.

Step 4: Factor the quadratic equation

We factor the quadratic equation $t^2 - 3t - 4 = 0$. We need two numbers that multiply to $-4$ and add up to $-3$. These numbers are $-4$ and $1$.

Step 5: Solve for t

From the factored equation $(t-4)(t+1) = 0$, we set each factor equal to zero to find the possible values for $t$. This gives us $t=4$ and $t=-1$.

Step 6: Select the valid time

Since time cannot be negative in this physical context, we discard $t=-1$ seconds. Therefore, the ball hits the ground after $4$ seconds.

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