Uses for formulas for sum and product of quadratic roots

[symbolipoint]

Are there practical uses for the formulas for the sum and product of quadratic roots? I have only seen the topic for these sum and product formulas in one section of any college algebra and intermediate algebra books, and then nothing more. I'm just curious if people, ... scientists or engineers, actually do anything PRACTICAL with these formulas, or are they just a topic in the textbooks with no further use? I never used these formulas in anything outside of just the mathematical, algebra exercises.

 

[Charles Link]

The sum of the roots and product of the roots I think is used most often in factoring something like $x^2+5x+6=(x+3)(x+2)=0$ I don't know that the actual $-b/a$ and $c/a$ is used that often, but 5 being the sum of the factors (i.e. of the additive constants) and 6 being the product is used quite extensively. Also, with quadratic equations, I have found the process of completing the square is extremely important and comes to use in many areas of both math and physics. $\\$ Editing... The student should also start to recognize the relationship between these additive constants in the factoring formulas, and the more formal way of factoring by using $(x-r_1)(x-r_2)=0$ where $r_1=\frac{-b+\sqrt{b^2-4ac}}{2a}$ and $r_2= \frac{-b-\sqrt{b^2-4ac}}{2a}$. All-in-all, the instruction of the quadratic forms is extremely important, and for anyone who is going to do mathematics, physics, or engineering, they do need a solid background in this subject. The sum $r_1+r_2=-b/a$ and product $r_1 r_2=c/a$ follows immediately from a little algebra.

 

[symbolipoint]

The sum of the roots and product of the roots I think is used most often in factoring something like $x^2+5x+6=(x+3)(x+2)=0$ I don't know that the actual $-b/a$ and $c/a$ is used that often, but 5 being the sum of the factors (i.e. of the additive constants) and 6 being the product is used quite extensively. Also, with quadratic equations, I have found the process of completing the square is extremely important and comes to use in many areas of both math and physics.

Is that all? Just an academic exercise, and that technical people in the real world do not use the sum and difference formula? If that were all, then most people with just "introductory algebra" knowledge already have enough knowledge and skill. You give a nice example, and it is something 'we' already understand, so not really a need for the sum of the roots formula and the product of the roots formula.

What if an engineer of whatever kind you like, in the real world has some ax^2+bx+c=0, for whatever variable x, and whatever actual real coefficients a, b, c, not necessarily rational coefficients. Would this engineer have a desire or practical use for the sum and difference formulas for the roots of the equation?

 

[Charles Link]

Is that all? Just an academic exercise, and that technical people in the real world do not use the sum and difference formula? If that were all, then most people with just "introductory algebra" knowledge already have enough knowledge and skill. You give a nice example, and it is something 'we' already understand, so not really a need for the sum of the roots formula and the product of the roots formula.

What if an engineer of whatever kind you like, in the real world has some ax^2+bx+c=0, for whatever variable x, and whatever actual real coefficients a, b, c, not necessarily rational coefficients. Would this engineer have a desire or practical use for the sum and difference formulas for the roots of the equation?

See the additional comments in my edited post above.

 

[symbolipoint]

See the additional comments in my edited post above.

Thanks about that. You are right that just a small amount of algebra skill is enough to derive those sum and difference formulas. Being shown how the relationships work is a good part of algebra instruction. I am still interested in whatever practical uses are either common, or that people have applied for practical purposes.

 

[mpresic]

Analyzing projectile motion, collisions in two dimensions, coupled mode problems are three examples that come to mind where it is important to be ale to solve and know the characteristics of solutions to quadratic equations. I worked professionally in organizations/laboratories for several decades, and there have been time when it was important to solve these equations. Once I was asked to solve a quadratic equation in a application by a superior. I presume he once knew how to solve one, and he forgot the method.

Police Forensics (projectile motion and collisions), and electric circuit and mechanical structure analysis (coupled modes) are two possibilities where the solutions have to be solved in real life.

The fact is almost everything I learned in high school mathematics is used later on.

 

[symbolipoint]

mpresic, I believe you. Quadratic equations are created and solved in the real working world. what I am asking is something else.

We know about a(x-s)(x-r)=0 and the formulas for s+r=-b/a and for sr=c/a.
My question is, do scientists and engineers USE THIS in their work? Are THESE FORMULAS regarding the roots only academic exercises and curiosities?

 

[fresh_42]

My question is, do scientists [ and engineers ] USE THIS in their work?

Scientists, yes, because they are coefficients of minimal polynomials, e.g. of field extensions: determinant (norm) and trace. Both important invariants.

... and engineers USE THIS in their work?

Hard to tell. Does anybody uses the fact, that $\pi$ is transcendental? That neutrinos can change flavor? That Milky Way and Andromeda will merge? At least Vieta's formulas are far more useful than the latter. E.g. they show that the coefficients depend continuously on the zeros which might be important in some systems of static engineering. In the end they are a very short way to solve quadratic equations and therewith useful, since this is quite often needed.

Are THESE FORMULAS regarding the roots only academic exercises and curiosities?

See above: determinate and trace.

 

[puzzled fish]

Are there practical uses for the formulas for the sum and product of quadratic roots?

Yes, for equations >= 2nd degree they are the origin of the Group of Substitutions and the foundation of modern Group Theory, historically developed first by Lagrange, then by Ruffini, Abel and Galois.

 

[symbolipoint]

Yes, for equations >= 2nd degree they are the origin of the Group of Substitutions and the foundation of modern Group Theory, historically developed first by Lagrange, then by Ruffini, Abel and Galois.

Fine. I would not argue against it. In all of my (maybe limited) experiences, I typically only applied either Completing-the-square, or directly used the general solution formula to find any "roots" of a quadratic equation, to solve a quadratic equation. Problems encoded from any lab situation or any problems from written documentation such as textbooks, or exams, or quizes, in certain cases yielded quadratic equations, and solutions for them ("roots") were needing to be solved for. Depending on the desired results, either general solution formula, or completing the square were all that were needed once having the equation. The formulas for sum and product of the roots? Never needed them. But most of my own uses or "applications" of quadratic equations were in learning Intermed algebra through Calc 3, and several physical science courses.