Qhov sib txawv ntawm kab zauv sib npaug thiab plaub qhov sib npaug

2024 Tus sau: Alex Aldridge | [email protected]. Kawg hloov kho: 2023-12-17 13:40

Linear Equation vs Quadratic Equation

Nyob hauv lej, algebraic equations yog equations uas yog tsim siv polynomials. Thaum sau meej meej cov kab zauv yuav yog ntawm daim ntawv P(x)=0, qhov twg x yog ib tug vector ntawm n tsis paub txawv txawv thiab P yog ib tug polynomial. Piv txwv li, P(x, y)=x⁴ + y³ + x²y + 5=0 yog ib qho kev sib npaug ntawm algebraic ntawm ob qhov sib txawv uas sau meej meej. Tsis tas li ntawd, (x+y)³=3x²y – 3zy⁴ yog ib qho kev sib npaug ntawm algebraic, tab sis nyob rau hauv daim ntawv implicit. Nws yuav siv daim ntawv Q(x, y, z)=x³ + y³ + 3xy² +3zy^{4=0, ib zaug sau meej.}

Tus yam ntxwv tseem ceeb ntawm kev sib npaug ntawm algebraic yog nws qib. Nws txhais tau tias yog lub zog siab tshaj plaws ntawm cov ntsiab lus tshwm sim hauv qhov sib npaug. Yog hais tias ib lub sij hawm muaj ob los yog ntau qhov sib txawv, qhov sib npaug ntawm cov exponents ntawm txhua qhov sib txawv yuav raug coj los ua lub zog ntawm lub sij hawm. Ua raws li qhov txhais no P(x, y)=0 yog degree 4 thaum Q(x, y, z)=0 yog degree 5.

Cov kab zauv sib npaug thiab cov kab zauv sib npaug yog ob hom kev sib npaug ntawm cov lej sib txawv. Qhov degree ntawm qhov sib npaug yog qhov sib txawv ntawm qhov sib npaug ntawm cov lej sib npaug.

Dab tsi yog linear equation?

A linear equation is an algebraic equation of degree 1. Piv txwv li, 4x + 5=0 is a linear equation of one variable. x + y + 5z=0 thiab 4x=3w + 5y + 7z yog linear equations ntawm 3 thiab 4 variables feem. Feem ntau, ib qho kab zauv sib npaug ntawm n qhov sib txawv yuav siv daim ntawv m₁x₁+m 2_x2_{+…+ m}n-1_x n-1_{+ m}n_xn _{=b. Ntawm no, x}i_{'s yog cov tsis paub hloov pauv, m}i_{'s thiab b yog tus lej tiag uas txhua tus ntawm m}i _{tsis yog xoom.}

Qhov kev sib npaug no sawv cev rau lub dav hlau siab hauv n-dimensional Euclidean qhov chaw. Tshwj xeeb, ob qhov sib npaug sib npaug sib npaug sawv cev rau txoj kab ncaj nraim hauv Cartesian dav hlau thiab peb qhov sib npaug sib npaug sib npaug sawv cev rau lub dav hlau ntawm Euclidean 3-space.

Dab tsi yog qhov sib npaug plaub npaug?

Ib qho kev sib npaug plaub yog ib qho kev sib npaug ntawm qhov thib ob. x² + 3x + 2=0 yog ib qho kev sib npaug sib npaug sib npaug. x² + y² + 3x=4 thiab 4x² + y²+ 2z^{2 + x + y + z=4 yog piv txwv ntawm kev sib npaug plaub npaug ntawm 2 thiab 3 qhov sib txawv raws li.}

Nyob rau hauv ib rooj plaub sib txawv, daim ntawv dav dav ntawm kev sib npaug plaub yog ax² + bx + c=0. Qhov twg a, b, c yog cov lej tiag tiag tawm ntawm qhov twg 'a' tsis yog xoom. Qhov kev ntxub ntxaug ∆=(b^{2 – 4ac) txiav txim siab qhov xwm txheej ntawm cov hauv paus hniav ntawm qhov sib npaug. Cov hauv paus hniav ntawm kab zauv yuav txawv tiag tiag, zoo sib xws thiab complex raws li ∆ yog qhov zoo, xoom thiab tsis zoo. Cov hauv paus hniav ntawm qhov sib npaug tuaj yeem pom tau yooj yim siv cov qauv x=(- b ± √∆) / 2a.}

Hauv ob qhov sib txawv, daim ntawv dav dav yuav yog ax² + by^{2 + cxy + dx + ex + f=0, thiab qhov no sawv cev rau lub conic (parabola, hyperbola lossis ellipse) hauv Cartesian dav hlau. Hauv qhov loj dua, hom kab zauv no sawv cev rau qhov chaw hyper-surfaces hu ua quadrics.}

Dab tsi yog qhov sib txawv ntawm kab sib npaug thiab plaub npaug?

• Ib kab zauv sib npaug yog qhov sib npaug ntawm qib 1, qhov sib npaug plaub yog qhov sib npaug ntawm qib 2.

• Hauv n-dimensional Euclidean qhov chaw, qhov chaw daws teeb meem ntawm n-variable linear equation yog lub dav hlau siab thaum lub n-variable quadratic equation yog quadric nto.