Relevance of complex roots of a quadratic equation

Question

Let's say Amy is a stunt pilot, planning on doing a parabolic dive in an air show:

$$y = x^2 + 4x +5$$

She hopes to use this trajectory to dive close to the ground (the $x$-axis, height is the $y$-axis), and pull up just before crashing to impress everyone.

To make sure she won't die, she asks two mathematicians to ensure the lack of roots of her parabola.

The first one, Betty, arrives at $x = -2 \pm \sqrt{-1}$. She concludes that since there is no number that satisfies that statement, there is no root. Amy will never hit the ground, and everybody will be impressed.

The second one, Charlie, remembers imaginary numbers, and arrives instead at $x = -2 \pm i$. Desperate, she urges Amy to cancel the air show for she will die a horrible death when her plane reaches $x = -2 \pm i$.

What should Amy do?

Or, what's the point of complex roots?

I am always told that complex numbers are “true” numbers, just as much as negative numbers or fractions. Yet in this example it seems to me that complex numbers “don't exist”.

Answer 1

And now suppose that Amy wants to compute the dimensions of a rectangular field with $48$ km² of which she also know that the largest sides have two more kilometers than the shortest ones. This leads her to the equation $x(x+2)=48$, which has two solutions: $6$ and $-8$. And she will ignore the answer $-8$, since, although $-8$ is indeed a number, it cannot possibly be a length, which is what she's after.

A similar situation occurs with the solutions $-2\pm i$. They are numbers, but these numbers have no meaning in this specific situation.

Answer 2

I guess there is logical problem in your real life situation you described there. The parabola you defined by $y=x^2+4x+5$, or to put it in another form $y=(x+2)^2+1$ is function which gots no real roots since it gots a vertex at the point $(-2,1)$.Furthermore the two ways you wrote down the roots of the equation, $x_{1,2}=-2\pm\sqrt{-1}$ and $x_{1,2}=-2\pm i$, both stand for the same number.

That means Amy just flips the airplane around at the point $(-2,1)$. When you consider only real roots there are 3 cases for quadratic equations: 2 dsitinctive roots, one doubled root and no real roots.

Only THESE cases are relevant for a real life situation like you described. But to go further in a mathematical context complex roots are more valuable. Without them the Fundamental Theorem of Algebra, which guarentees the existence of - at least - roots for any polynomial, would not work out.

Answer 3

Amy should go for Betty.

In the case you describe the reality is translated to a mathematical model.

Then Betty applies some mathematics in that model and reaches conclusions, and finally a converse translation of conclusions takes place from model to reality.

If this translation cannot take place then something is wrong with the model.

E.g. it can be taken "too wide".

This can be said of the model of Charlie.

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This is mainly how it works between math and reality. First the (inspiring) links with reality are somehow "modded out", then (abstract) mathematics takes place leading to conclusions and finally these are projected on reality again.

Answer 4

Amy should find a real (pun intended) mathematician, like for instance Mabel, who knows a little calculus. What Mabel would say is this:

If the trajectory

$y(x) = x^2 + 4x + 5 \tag 1$

attains a minimum value of height $y_m$, it occurs at that value $x_m$ of $x$ where

$y'(x) = 2x + 4 = 0; \tag 2$

that is, at

$x = x_m = -2; \tag 3$

Mabel knows $x_m$ is in fact a minimum of $y(x)$ since she evaluates

$y''(x) = 2 > 0 \; \text{for all} \; x; \tag 4$

in particular,

$y''(x_m) = 2 > 0, \tag 5$

so Mabel concludes that $y(x_m)$ the smallest possible height the plane will attain if it adheres rigorously to the dive plan $y(x)$; with just a tad more arithmetic, Mabel finds that

$y_m = y(x_m) = y(-2) = (-2)^2 + 4(-2) + 5 = 4 - 8 + 5 = 1; \tag 6$

so Mabel confidently tells Amy that her plane won't crash, since it will never fly lower than $1$ unit of height above the ground.

After living through her successful dive, Amy then decides to return to her local community college for a refresher course in basic calculus, where she comes to understand it is $x_m$ which really matters here, not the zeroes $-2 \pm i$ of $y(x)$.