Quadratic equations that represent situations cannot always be neatly put into factored form or easily solved by finding square roots. Completing the square is a workable strategy, but for some equations, it may involve many cumbersome steps. Graphing is also a handy way to solve the equations, but it doesn’t always give us precise solutions.

With the quadratic formula, we can solve these equations more readily and precisely.

Here’s an example: Function $h$ models the height of an object, in meters, $t$ seconds after it is launched into the air. It is is defined by $h(t)=\text-5t^2+25t$.

To know how much time it would take the object to reach 15 meters, we could solve the equation $15=\text-5t^2+25t$. How should we do it?

• Rewriting it in standard form gives $\text-5t^2+25t-15=0$. The expression on the left side of the equation cannot be written in factored form, however.
• Completing the square isn't convenient because the coefficient of the squared term is not a perfect square and the coefficient the linear term is an odd number.
• Let’s use the quadratic formula, using $a=\text-5,b=25,\text{and }c=\text-15$!

The solutions tell us that there are two times after the launch when the object is at a height of 15 meters: at about 0.7 second (as the object is going up) and 4.3 seconds (as it comes back down).