Chemist use the quadratic equation all the time to find concentrations in equilibrium reactions.
Mechanics in physics, kinematics to be precise, use this formula to find position, time velocity or acceleration of moving objects.
X = Vot + 1/2at2
--------------------------------a quadratic formula
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By using the quadratic equation formula or by completing the square
Basically, we use mathematics in 'real life' to make real life more understandable. To this, we would use quadratic equations to model processes that (at least to a certain degree) resemble reality. One such example of a quadratic equation is v^2 = u^2 + 2as. This formula determines the final speed that you are going based on a constant acceleration. The reason we are able to use a quadratic equation here is because the cars force is relatively constant, and F=ma so the acceleration is constant. However, because drag has a large impact on forces at high speeds, this equation is not suitable for those speeds. For interest's sake, the drag formula states that Force is proportional to velocity^3.
The question is based on the false assumption that the quadratic formula is not used in daily life. Wrong, it IS!
As you probably suspect, there are no non-mathematical situations in which you would use the quadratic formula.
To find the roots (solutions) of a quadratic equation.