# Quadratic, cubic, quartic, quintic equations

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• fog37
In summary, the terms "linear, quadratic, cubic, and quartic equation" refer to equations with one variable, while the term "quadratic equation" can also be applied to equations with two variables if specified. The expression "ax^2 + by^2 + cxy + dx + ey + f = 0" is the correct form for a quadratic equation in three variables, and the term "elliptic equation" may also be used to describe this type of equation. The solutions of these equations are the roots, or y-intercepts, in the plane. The usage of the term "quadratic" is context-sensitive and may refer to an expression with one or both variables raised to the 2nd power.
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.

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

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

symbolipoint
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...

fog37 said:
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.
By the way, the form
fog37 said:
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

Last edited:
fog37 said:
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.

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?

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...

fog37 said:
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.

fog37 said:
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.

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##

fog37 said:
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.##

fresh_42 said:
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:
Regards,
Buzz

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.

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

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##.

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

## 1. What are quadratic equations?

Quadratic equations are algebraic expressions with a degree of 2, meaning that the highest exponent of the variable is 2. They are typically written in the form of ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable.

## 2. How do you solve quadratic equations?

There are several methods for solving quadratic equations, including factoring, completing the square, and using the quadratic formula. The most commonly used method is the quadratic formula, which is: x = (-b ± √(b^2 - 4ac)) / 2a.

## 3. What is the difference between cubic and quartic equations?

The main difference between cubic and quartic equations is the degree of the polynomial. Cubic equations have a degree of 3, while quartic equations have a degree of 4. This means that quartic equations can have up to 4 solutions, while cubic equations can have up to 3 solutions.

## 4. Can quintic equations be solved algebraically?

No, quintic equations cannot be solved algebraically using the methods we use for quadratic, cubic, and quartic equations. This is because there is no general formula for finding the solutions of a quintic equation. Instead, numerical methods must be used to approximate the solutions.

## 5. What are some real-life applications of quadratic, cubic, and quartic equations?

Quadratic, cubic, and quartic equations are used in many fields of science and engineering, including physics, chemistry, and economics. They can be used to model and solve problems related to motion, growth and decay, and optimization. For example, the motion of a projectile can be described using a quadratic equation, and the growth of a population can be modeled using a cubic or quartic equation.

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