Quadratic equation

The quadratic equation (にじほうていしき, quadratic equation) is an equation to describe the point zero of the second multinomial expression with kind of the equation. It is the equation about the n variable multinomial expression generally

${\displaystyle 0=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{i,j}x_{i}x_{j}+\sum _{k=1}^{n}b_{k}x_{k}+c}$

I call (but, in a_i,j, b_k, c, a_i,j + a_j,i of fixed number and at least one is not 0) n variable quadratic equation, n former quadratic equation. The geometric consideration for the point zero meeting is carried out about these historically (particularly about the thing of the real number coefficient) and is well known (you should refer to a conicoid for the general multivariate quadratic equation in a conic section about the 2 yuan quadratic equation).

Speaking of merely quadratic equation, it is the quadratic equation of 1 variable in secondary education of Japan

${\displaystyle ax^{2}+bx+c=0}$

I often point to (a ≠ 0 that a real number constant stands out in a, b, c). I describe this in this clause as follows.

Table of contents

History

Mathematician フワーリズミー of Islam discovered that there were two roots in the quadratic equation in the Middle Ages when I played an active part for Abbasid dynasty Era. "Al = フワーリズミー (translation into Latin "Al gobiid Tomi デ gnu Melo インドルム (Algoritmi de numero Indorum)") was translated into Latin, and it came to Europe about a number of India" writing of フワーリズミー. フワーリズミー expressed the unknown quantity in the quadratic equation by the word "shay'" (shy = and a certain thing), but, for the stage when writing of フワーリズミー is introduced into Europe, "x" is said when I pass "sh" (シ) and Portuguese to read when "sh" of shay' was replaced with "x". It is said that such a background is in calling an unknown thing "x". ^[1]

Definition

The quadratic equation is an algebraic equation of degree 2. According to a definition,

(*) ${\displaystyle ax^{2}+bx+c=0}$ 

I am referred (in a ≠ 0, b, c the fixed number). The general form (generalized form) of the quadratic equation means this. Furthermore, about a quadratic equation, special form with some characteristics is thought about. I use the following terms in this clause conveniently.

A second clause coefficient is 1 equation

(**) ${\displaystyle x^{2}+px+q=0}$ 

I call (in p, q the fixed number) the regular form (normal form) of second 整方程式 or the quadratic equation. I divide statement of equation (*) of the general form by a ≠ 0 and can be going to have regular form (normalization):

(***) ${\displaystyle p={\frac {b}{a}},\ q={\frac {c}{a}}.}$ 

Still further, equation in an appearance without the primary clause

${\displaystyle k(x+l)^{2}+m=0}$ 

I call (in k ≠ 0, l, m the fixed number) the canonical form (standard form) of the quadratic equation. This is t = x + with a variable l It is と equation kt2 + m = 0 not to really have a primary clause about t if I convert it.

Square completion

In a quadratic equation${\displaystyle \left(x-\alpha \right)^{2}}$ Transformation to create clauses of the の form is called square completion (basic transformation).

In the body which is not 標数 2, I can do equation (**) of the regular form in a canonical form by square completion. Quadratic equation

${\displaystyle x^{2}+px+q}$ 

に vs. do it, and is official${\displaystyle \left(x+y\right)^{2}=x^{2}+2xy+y^{2}}$ To use を

${\displaystyle x^{2}+2\left({\frac {p}{2}}\right)x+q}$ 

In being rich, and doing, and adding (p/2)², and going down

${\displaystyle \left x^{2}+2\left({\frac {p}{2}}\right)x+\left({\frac {p}{2}}\right)^{2}\right +\left -\left({\frac {p}{2}}\right)^{2}+q\right}$ 

としてからまとめると

${\displaystyle \left(x+{\frac {p}{2}}\right)^{2}-\left({\frac {p^{2}}{4}}-q\right)}$ 

となる. Therefore,${\displaystyle t=x+{\frac {p}{2}},m={\frac {p^{2}}{4}}-q}$  When I put と

${\displaystyle x^{2}+px+q=0\iff t^{2}-m=0}$ 

The neighbor, the equation of the canonical form about variable t are provided.

The technique of the square completion is used for the standardization of the conic section besides.

Root of the quadratic equation

• The root of normalized canonical form quadratic equation x2 - m = 0 is called the square root of m. I may express one root picked optionally in √ m. The square root of root of x2 +1 = 0 namely -1 is called in particular an imaginary unit.
• Relations of a root and the coefficient： When a root of equation (**) of the regular form is α, β
  • ${\displaystyle -p=\alpha +\beta ,\ q=\alpha \beta}.$ 
• In the general ring which is not a body and a domain, the number of roots of the quadratic equation may not be two.
• I call an irrational number to become the root of the quadratic equation of the rational number coefficient the second irrational number. The body which added the second irrational number in several Yuri is called the second body.

Official of the solution

The details refer to "the formula of the solution of the quadratic equation"

I do the quadratic equation as follows and can untie it generally. Generally this is used with the body which is not 標数 2. For quadratic equation ax2 + bx + c = 0,

${\displaystyle {\begin{aligned}a\!\left(x+{\frac {b}{2a}}\right)^{\!2}+c-{\frac {b^{2}}{4a}}=0&\iff \left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}-4ac}{4a^{2}}}\\&\iff x+{\frac {b}{2a}}=\pm {\frac {\sqrt {b^{2}-4ac}}{2a}}\\&\iff x={\cfrac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}\end{aligned}}}$ 

I get を. This is the formula of the solution of the quadratic equation. Furthermore, if b = 2b' is far

${\displaystyle x={\frac {-b'\pm {\sqrt {(b')^{2}-ac}}}{a}}}$ 

I can write と.

Quadratic equation of the real number coefficient

An imaginary unit and complex number

Quadratic equation of the real number coefficient

${\displaystyle x^{2}+1=0}$ 

Oh, I do not have a solution in the range of a real number. All real number coefficient quadratic equations come to have a root by introducing new cordage i which is not a real number as the cause.

I call the number that I can write as x + i y using real number x, y complex number. The algebraic equation of all complex number coefficients is shown to have a root in the range of complex number by all means (algebraic basic theorem).

The second discriminant

For an algebraic equation, call flat method Δ ² of difference product Δ made from the whole root with discriminant and can distinguish it whether a multinomial expression has an equal root. The difference product consisting of roots of second multinomial expression ax2 + bx + c = a(x-α)(x-β) (a ≠ 0) is this case, difference Δ = β - α; discriminant Δ ² = (β-α) ² by the relations of a root and the coefficient as a rational expression of coefficient a, b, c

${\displaystyle \Delta ^{2}={\frac {b^{2}-4ac}{a^{2}}}}$ 

I am given で. When use this Δ ²; the formula of the root of the quadratic equation

${\displaystyle x={\frac {-b}{2a}}\pm {\frac {\sqrt {\Delta ^{2}}}{2}}}$ 

I can distinguish it using this discriminant Δ ² to be able to write と whether it is solved in the range of a real number for the quadratic equation of the real number coefficient as a particularity in case of the second (with the result that is similar in the case of the cubic equation).

According to a mark as for the judgment of the kind of the root by discriminant Δ ²

• Δ ² > In the case of 0: 2 different real roots
• In the case of Δ ² = 0: Equal root of real number
• In the case of Δ ² <0: 2 different imaginary roots

I can speak として. Because square a² of real number a which is not 0 is always plus, here, the decision of the mark of discriminant Δ ² is attributed to deciding a mark of molecular D = b²-4ac. Thus, I often call it the discriminant of the quadratic equation with D. Because it becomes Δ ² = D = p2-4q for regular form x2 + px + q = 0, in the first place I do not need distinction.

Source

1. ^ http://www.aii-t.org/j/maqha/thaqafa/arqam.htm

Allied item

• Continued fraction
• The golden mean
• Drawing figures with a ruler and compasses

It is acquired by "https://ja.wikipedia.org/w/index.php?title= quadratic equation &oldid=61422481"

This article is taken from the Japanese Wikipedia Quadratic equation

This article is distributed by cc-by-sa or GFDL license in accordance with the provisions of Wikipedia.

Wikipedia and Tranpedia does not guarantee the accuracy of this document. See our disclaimer for more information.

In addition, Tranpedia is simply not responsible for any show is only by translating the writings of foreign licenses that are compatible with CC-BY-SA license information.