Quadratic, cubic, quartic, quintic equations

[fog37]

Hello,
In general, any equation is a statement of equality between two expressions. In the 2D case, equations generally involve the two variables $x$ and $y$ or either variable alone if we require the other variable to be equal to zero.
The most general quadratic equation should be the polynomial $$ay^2+by+cx^2+dx+e=0$$ where both variables are raised to the 2nd power. Why instead, is the polynomial of the form $y=ax^2+bx+c$ presented in introductory textbooks as the stereotypical and general quadratic equation?

If we set $y=0$, we get what is called the "quadratic equation": $$ax^2+bx+c=0$$This quadratic equation has only one variable, $x$, raised to the 2nd power and is solvable using the quadratic formula to find the two solutions, called roots, $(x_1,0)$ and $(x_2,0)$, which are essentially the -intercepts.

So when I hear that only linear, quadratic, cubic and quartic equations have methods to find their solutions (there are no methods for quintic equations), does it means that there are methods to find the roots of linear, quadratic, cubic and quartic equations involving only the $x$ variable? For example, the solutions of $0=ax^3+bx^2+cx+d$ but not the solutions of equation $y=ax^3+bx^2+cx+d$.

Thanks.

 

[scottdave]

When you have 2 variables, let's just look at a simple line like y = 2x + 3. There is an infinite number of points which are on this line and satisfy the equation. The same can be said for the higher order equations with 2 or more variables

 

[fresh_42]

If it is said "a linear, quadratic, cubic and quartic equation", then it is meant to have only one variable.

 

[fog37]

Thank you.

Fresh_42, you mention that a linear, quadratic, cubic and quartic equation is an equation with only one variable. So how do we call an expression like $$y=x^2+x+2$$?
Isn't it supposed to be called quadratic equation as well? We call an equation with two variables $y=4x+5$ a linear equation...

 

[Buzz Bloom]

Why instead, is the polynomial of the form y=ax^2+bx+c presented in introductory textbooks as the stereotypical and general quadratic equation?

Hi foq:

You are asking why a term used in mathematics is what it is. In general, it is the same reason for why a word in a spoken language means what it means. The reason is the word's or term's etymology, that is the history of changes in usage over long periods of time. I do not know the etymology of the term "quadratic" except for the short explanations given online, such as.

https://www.etymonline.com/word/quadratic
https://en.wiktionary.org/wiki/quadratic​

By the way, the form

y=ax2+bx+c

is technically not a quadratic equation, but if you assume the intent is to solve for x when y is specified, the equation can be rewritten as

0=ax^2+bx+(c-y)
​

You also have some further confusion about the higher degree equations.

The usage of "quadratic can be applied to equations with more than one variable if the term is used specifically with the specification of the number of variables. For example: "The quadratic equation in two variables."

BTW, the form you have for this in your first equation is wrong.The correct form is:

ax^2 + by^2 + cxy + dx + ey + f = 0​

I suggest you may want to (as a homework exercise) figure out the correct form for a quadratic equation in three variables.

Regards,
Buzz

 

[fresh_42]

Thank you.

Fresh_42, you mention that a linear, quadratic, cubic and quartic equation is an equation with only one variable. So how do we call an expression like $$y=x^2+x+2$$?
Isn't it supposed to be called quadratic equation as well? We call an equation with two variables $y=4x+5$ a linear equation...

I would call it an elliptic equation. But you may call it quadratic as well. It only shows, that the notion is context sensitive. Equation is the problematic term here and should be polynomial, in which case it is clear that one has to add in how many variables.

 

[fog37]

Hi Buzz,
You are right: ax^2 + by^2 + cxy + dx + ey + f = 0 is the correct form (I was missing the cross term).

I guess "quadratic" means an equation where one or both of the involved variables are raised to the 2nd power. Is that not correct?

 

[fog37]

I thought that if an mathematical expression has an equal sign it must be an equation. Polynomial means there are multiple addends in the expression...

I never heard that expression being called "elliptic". I will read about it...

 

[fresh_42]

Hi Buzz,
You are right: ax^2 + by^2 + cxy + dx + ey + f = 0 is the correct form (I was missing the cross term).

I guess "quadratic" means an equation where one or both of the involved variables are raised to the 2nd power. Is that not correct?

No really. Quadratic is usually only used in case of one variable. Otherwise the normal wording is of degree two. You can refer to a quadratic curve or hyperplane, in which case the second variable is hidden in curve, resp. hyperplane.

 

[fresh_42]

I never heard that expression being called "elliptic". I will read about it..

You're right, an elliptic curve is a bit different: https://en.wikipedia.org/wiki/Elliptic_curve

 

[fresh_42]

I thought that if an mathematical expression has an equal sign it must be an equation.

Sure, but this covers all equations, polynomial or not. Quadratic plus equation is the problem as it does not mention quadratic in what?
In this sense, $y=\sin^2x$ would be quadratic as well. So why do you imply a multivariate polynomial but not a trigonometric function? It's simply underdetermined which is why requires additional information, either by context or explicitly.

 

[fog37]

Thank you. you are right, the expression does not have to be a polynomial...

I see, so quadratic equation only refers to an equation with only one variable, same goes for quintic, etc. So the solutions are always the roots, i.e. the y-intercepts, in the plane... But for the linear case, we have no problem calling linear equation an expression with two variables $y=2x+1$

 

[fresh_42]

Thank you. you are right, the expression does not have to be a polynomial...

I see, so quadratic equation only refers to an equation with only one variable, same goes for quintic, etc. So the solutions are always the roots, i.e. the y-intercepts, in the plane... But for the linear case, we have no problem calling linear equation an expression with two variables $y=2x+1$

Again: context! In this context, linear refers to the shape of the graph. Mathematically it is affine linear as $(0,0)$ isn't part of the graph, but still linear, as the dependency of $y$ form $x$ is meant.

Linear in linear solvability refers to the degree of $2x+1$ and solvability to $2x+1=0$. Linear equation is underdetermined as it doesn't qualify linearity in what? Usually the context answers this question so there is no need to specify it further, but as a standalone notion it is not sufficient, resp. the most easy one is assumed: $2x+1=0.$

 

[Buzz Bloom]

Quadratic is usually only used in case of one variable. Otherwise the normal wording is of degree two.

Hi fresh:

https://en.wikipedia.org/wiki/Equation

An algebraic equation or polynomial equation is an equation in which both sides are polynomials (see also system of polynomial equations). These are further classified by degree:

• linear equation for degree one
• quadratic equation for degree two
• cubic equation for degree three
• quartic equation for degree four
• quintic equation for degree five
• sextic equation for degree six
• septic equation for degree seven
• octic equation for degree eight

Regards,
Buzz

 

[fresh_42]

No,

In mathematics, an equation is a statement of an equality containing one or more variables.

And even this is wrong.

An equation is what sets two different expressions to be equal. How these expressions are composed is irrelevant. So everything that has a '$=$' sign in it is an equation. You can't say that $\dfrac{4}{2}=2$ isn't an equation. It's not on your list above. You can't say, that $1=\sin x$ isn't an equation. It's not on your list above. And what about roots, logarithms, exponential functions, etc.? Wiki calls it identities instead, but this is linguistic nitpicking. In the end, they are all equivalence relations, since obviously the two sides are not identical by written signs. So the question is, which equivalence relations do we call equation, which identity and which merely equivalent? This is for linguists and philosophers, not for mathematicians.

E.g. this is on the German Wikipedia page:

An equation in mathematics is a statement about the equality of two terms, which is symbolized with the help of the equal sign ("="). Formally, an equation has the form $T_ {1} = T_ {2},$ where the term $T_ {1}$ is the left side and the term $T_ {2}$ is the right side of the equation.

Now are American equations different from German ones, or shall we start a controversy which one is correct? The names are either determined by historical usances or context sensitive.

If you insist on a proper definition, we need a proper reference. Wikipedia is none. This one is:
https://ncatlab.org/nlab/show/equation

 

[fog37]

Fair corrections. I should have not been sloppy on the definition of an equation.
Focusing on algebraic, i.e. polynomial, equations, Wikipedia mentions (see Buzz Bloom reply) the name of polynomial equations of various degree (linear, quadratic, cubic, etc.) and all these equations have only one variable, the variable $x$. For example, $y=ax^3+bx^2+cx+d$ is called the cubic function and when $y=0$, it becomes the cubic equation $ax^3+bx^2+cx+d=0$. So I guess a hypothetical equation that involves both variables $x$ and $y$, like $y^2+x^3+2x^2+x+5=0$ should not to be called a cubic equation just because the highest monomial degree is 3 and because the name cubic equation is reserved for $ax^3+bx^2+cx+d=0$.

 

[fresh_42]

$y^2+x^3+2x^2+x+5=0$ is a polynomial equation in two variables of degree three.