Quadratic Equations in Real Life

All uses, applications, and benefits of quadratic equations in “real life” can be roughly split into two types:

So why the hell do we even need these “puzzles” if they have no practical use? They do have a use, just mostly for you personally — they are solid, varied training for your ability to turn text and sneaky conditions into formulas. And overall, even these puzzles show that quadratic equations are not some detached nonsense floating away from reality. They do show up in real situations, just usually only if the conditions are chosen cunningly.

Quadratic equations hide in daily life damn near everywhere: cardboard boxes, walkways, shopping trips, and so on… Yeah, most of these problems are still “puzzles”. But there are also some nice exceptions — situations from actual life that naturally reduce to quadratic equations.

A lot of people hate physics, and that is a shame. Physics gives you one of the biggest piles of examples of situations and problems that reduce to quadratic equations. And you get both kinds here: genuinely real problems and the usual cleverly staged “puzzles”.

When forces act on objects, their speed changes: a soccer ball kicked by a foot, a snowball thrown by hand, a bow string launching an arrow, or Earth’s gravity speeding up a hammer falling downward. This change in speed can be measured. “The speed at which speed changes” sounds weird, but that quantity is called acceleration and is denoted by the letter a.

The good news is that simple types of accelerated motion are described directly by quadratic equations. No tricky setup is needed at all. The classic example is free fall under Earth’s gravity. Our planet gives any object an extra acceleration of g = 10 meters per second for every second it stays in the air. If an object is dropped from height $h_0$ with initial speed $v_0$, then its height h at time t is given by:

The sign in front of $v_0$ depends on the direction of the initial velocity. If the object starts downward, you use a minus. If it starts upward, you use a plus.

Quadratic equations have also crawled into electric circuits. Circuit elements can be connected in series or in parallel, which lets you flexibly control voltage, current, and resistance. For parallel connections, the total resistance $R_t$ is calculated from the individual resistances $R_1$, $R_2$, and so on by the lovely ugly formula:

You can invent a stupid amount of puzzle problems from motion at constant speed. Even people who hate physics know the simple formula $S = V \cdot t$, which links distance, speed, and time. It is pretty intuitive that if you sprint to the store for chocolate at V = 2 meters per second, then in t = 10 seconds you will cover $S = V \cdot t = 2 \cdot 10 = 20$ meters.

From that basic distance formula, you can derive formulas for speed and time by simple transformations. That gives the famous school trio:

The classical constant-speed formula S = Vt can be generalized to basically any problem where there is some kind of “work” in the everyday sense, done by someone or something at some rate over some time.

Distance S gets replaced by some “amount of work” W, whatever units it may use: the area of a fence to paint, the volume of a tank to fill, the number of parts a factory has to make, and so on. Speed V becomes “productivity”, “output”, or just a work rate P: how many square meters get painted per hour, how many liters flow into a tank per minute, how many parts get made per day, and so on.

The formulas are exactly the same, only the letters change. For example, if a worker makes P = 8 parts per hour, then in t = 10 hours they make $8 \cdot 10 = 80 = W$ parts. That gives the main formula W = Pt, and from it you get the work-rate and work-time formulas, exactly like in constant-speed motion:

In chemistry, including everyday chemistry, in metallurgy, and even in plain old concrete mixing, you constantly run into situations where substances are combined. In English, mixture is the broad general word,solution is used when one substance is dissolved in another, and alloy is the standard word for a metallic mixture. So in this section we will look at a liquid solution, a metal alloy, and a gas mixture.

For all these systems, it is crucial to understand what fraction of the total mass or volume is made up by the substance you care about. For example, what fraction of an acid-and-water solution is acid. That ratio is called the concentration of the substance, say A, and it is denoted by c.

Concentration is usually written in percent, but in calculations you use decimals. For example, a hydrogen peroxide solution with concentration 3% means c = 0.03.

And of course, quadratic equations are used in mathematics itself. Plenty of fun questions in both algebra and geometry lead straight to them. Here are some examples from both areas.