Learning Quadratic Formula opens up a lot of math and applications

Learning the quadratic formula and the derivations and manipulations in it opens up a lot of math.    A quadratic is of the form

a+bx+cx^2

Quadratic forms appear in the following contexts.

1. The potential of a spring or harmonic oscillator.  This demonstrates force, potential, potential energy and conservation of energy.
2. A system of harmonic oscillators can be used to represent many physical systems.
3. The Gaussian or normal probability distribution is the exponential of a quadratic for the probability density.  This generalizes to multiple dimensions.
4. The quadratic interest rate model in finance.  With quadratics we can use x a random variable that can go negative, but x^2 and some other quadratic forms stay positive, so the interest rate can be the quadratic form instead of the variable.  This leads to formulas in finance.  See Beaglehole Tenney for the development of this approach.
5. In quantum mechanics we can use a quadratic potential and get solutions for the wave functions.  This is actually related math to the Beaglehole Tenney case mentioned above for interest rate models.
6. If we have a non-linear optimization, we can use a quadratic to model it in many cases.  For these, we can get solutions more easily.
7. Portfolio optimization in the simplest case is a quadratic optimization.    The variance of the portfolio is quadratic in the portfolio weights.  The expected return is linear.  One can maximize the expected return for a given level of the variance.
8. Over and over the cases that can be solved and worked with in practical problems will turn out to be quadratic.

High school students learn the quadratic formula.  They often are not taught to learn the derivation, but sometimes are.   Mastering the quadratic fully including the derivation of the quadratic formula should be one of the goals of Algebra II.

This should, ideally, include completing the quadratic form in the exponent for calculations involving the normal distribution.

This is math that is within the grasp and understanding of students who complete the common core algebra 2 standards.  These are required for graduation in most states.  This is doing math and doing these derivations and understanding them and their applications is thinking mathematically.  It is within the grasp and understanding of anyone who completes the common  core algebra II standards.