Oh that the youth would know the cubic, solve the cubic

This is a generic cubic:

0 = e x³ + f x² + g x + h

with coefficients e, f, g, and h.

This is the Van der Waal's EOS, a cubic:

P = [R T / (ν - b)] - [a / (ν²)]                           or,

P ν² = [R T ν² / (ν - b)] - a                              or,

P ν² (ν - b) = (R T ν²) - a (ν - b)                      or,

P ν³ - b P ν² = (R T ν²) - (a ν) + (a b)             or,

P ν³ - (b P + R T) ν² + (a ν) - (a b) = 0           where,

now,

e = P;

f = -(b P + R T);

g = a;

h = a b

and we're using ν as the independent variable instead of x, of course, but it's the same format.

[For the curious: P is pressure, R is the gas constant, T is absolute temperature, a and b are constants based on critical properties of the specific gas, and ν is specific volume.]

13/27 of my students know how to solve a cubic function.

That is less than half. You solve cubics in high school. Sometimes you solve them in middle school.

It's not surprising that many people criticize engineers for being out of touch with reality--so many just love to play with equations without thinking about what the equations are, or how they can be used/manipulated.

They are tools, not magic, and need to be treated with respect and that understanding, but not reverence.

During the final, one of my students said to me:

"No one ever taught me to solve an equation like this."

I thought about that. What I said was along the lines of:

If you don't know how to solve it, show me clearly, as far as you can, what you've done, and what you would do.

What I thought was:

No. Someone taught you how to solve a cubic; you've done it many times. You just don't recognize this is an instance of the cubic you know how to solve, because you view this equation as being special, because it is the Van der Waals EOS. It can be manipulated just like any other polynomial that you know.

Then I cried and cried and cried and probably vented to Twitter.