Lub keeb kwm ntawm ib quadratic equation: algebraic thiab duab lub ntsiab lus

Nyob rau hauv algebra square yog hu ua ib tug thib ob kev txiav txim equation. Los ntawm kab zauv hais xaam qhia, uas muaj nyob rau hauv nws cov muaj pes tsawg leeg ntawm ib los yog ntau tshaj tsis paub hais tias. Ob txhais-thiaj equation - ib tug zauv kab zauv muaj tsawg kawg yog ib tsis paub hais tias nyob rau hauv square degrees. Lub quadratic equation - thib ob-thiaj equation qhia yog leejtwg los yog sib npaug zos rau zero. Daws cov kab zauv square yog tib tug uas txiav txim seb lub square cov hauv paus hniav ntawm cov kab zauv. Raug quadratic equation nyob rau hauv lub dav dav daim ntawv:

W * c ^ 2 + T * c + O = 0

nyob rau W, T - lub coefficients ntawm cov keeb kwm ntawm lub quadratic equation;

O - free coefficient;

c - hauv paus ntawm lub quadratic equation (yeej ib txwm muaj ob tug tseem ceeb c1 thiab c2).

Raws li twb tau hais, qhov teeb meem ntawm kev daws rau ib tug quadratic equation - nrhiav cov keeb kwm ntawm ib quadratic equation. Yuav kom nrhiav tau lawv, koj yuav tsum nrhiav ib tug discriminant:

N = T ^ 2 - 4 * W * O

Lub discriminant qauv tsim nyog rau kev nrhiav cov ntsiab hauv paus c1 thiab c2:

c1 = (-t + √N) / 2 * W thiab c2 = (-t - √N) / 2 * W

Yog hais tias lub quadratic equation ntawm lub dav daim ntawv ceeb ntawm lub hauv paus ntawm T muaj ib tug ntau nqi, cov kab zauv hloov los ntawm:

W * c ^ 2 + 2 * U * c + O = 0

Thiab cov hauv paus hniav zoo li cov kev qhia:

c1 = [-u + √ (U ^ 2-W * O)] / W thiab c2 = [-u - √ (U ^ 2-W * O)] / W

Feem ntau kab zauv yuav muaj ib tug me ntsis txawv tsos thaum C_2 yuav tsis muaj coefficient W. Nyob rau hauv cov ntaub ntawv no, lub saum toj no kab zauv muaj rau hauv daim:

c ^ 2 + F * c + L = 0

qhov twg F - qhov zoo tshaj ntawm lub hauv paus;

L - free tshaj;

c - hauv paus ntawm lub square (yeej ib txwm muaj ob tug tseem ceeb c1 thiab c2).

Qhov no hom ntawm kab zauv hu ua ib tug quadratic equation muab. Lub npe "txo" mus los ntawm mis actuation raug quadratic equation, yog hais tias tus coefficient ntawm W paus muaj ib tug nqi ntawm ib tug. Nyob rau hauv cov ntaub ntawv no, cov hauv paus hniav ntawm cov quadratic equation:

c1 = -f / 2 + √ [(F / 2) ^ 2-L)] thiab c2 = -f / 2 - √ [(F / 2) ^ 2-L)]

Nyob rau hauv cov ntaub ntawv ntawm txawm tseem ceeb ntawm cov coefficient ntawm lub F paus hauv paus hniav yuav muaj ib tug daws:

c1 = -f + √ (F ^ 2-L) c2 = -f - √ (F ^ 2-L)

Yog hais tias peb tham txog quadratic equations, nws yog tsim nyog yuav tau nco qab lub theorem ntawm Vieta. Nws hais tias cov nram qab no kev cai rau cov txo quadratic equation:

c ^ 2 + F * c + L = 0

c1 + c2 = -f thiab c1 * c2 = L

Nyob rau hauv kev quadratic equation quadratic equation hauv paus hniav yog yam dependencies:

W * c ^ 2 + T * c + O = 0

c1 + c2 = -t / W thiab c1 * c2 = O / W

Tam sim no xav txog cov kev xaiv uas ntawm quadratic equations thiab lawv cov kev daws. Tag nrho cov ntawm lawv muaj peev xwm yuav ob tug, raws li yog ib tug tswv cuab ntawm C_2 yog uas ploj lawm, ces tus kab zauv yuav tsis tau square. yog li ntawd:

1. W * c ^ 2 + T * c = 0 ntawm lub quadratic equation embodiment tsis muaj free tshaj (neeg).

Cov tshuaj no yog:

W * c ^ 2 = -t * c

c1 = 0, c2 = -t / W

2. W * c ^ 2 + O = 0 ntawm lub quadratic equation embodiment tsis muaj tus thib ob lub sij hawm, thaum lub tib modulo cov keeb kwm ntawm lub quadratic equation.

Cov tshuaj no yog:

W * c ^ 2 = -O

c1 = √ (-O / W), c2 = - √ (-O / W)

Tag nrho cov no yog algebra. Xav txog cov geometric ntsiab lus ntawm uas muaj ib tug quadratic equation. tus thib ob kev txiav txim equation nyob rau hauv lub geometry yog piav los ntawm ib tug parabola muaj nuj nqi. heev feem ntau cov neeg ua hauj lwm yog los nrhiav cov keeb kwm ntawm ib quadratic equation rau high school cov tub ntxhais kawm? Cov hauv paus hniav muab lub tswvyim ntawm yuav ua li cas tshuam lub teeb muaj nuj nqi (parabola) nrog cov koom ua ke axis - lub kab rov tav. Yog hais tias, thaum nej tau txiav txim siab tus quadratic equation, peb tau txais lub irrational kev txiav txim siab ntawm cov keeb kwm, ces cov kev tshuam yuav tsis tau. Yog hais tias lub hauv paus muaj ib tug lub cev muaj nuj nqis, cov muaj nuj nqi crosses cov x-axis nyob rau hauv ib qho chaw. Yog hais tias ob cag, ces, ntsig txog, - ob lub ntsiab lus ntawm kev sib tshuam.

Nws yog ib nqi sau cia hais tias nyob rau hauv lub irrational cov hauv paus hniav hais ib tug tsis zoo nqi nyob rau hauv lub hauv paus, ntawm lub hauv paus nrhiav. Lub cev muaj nqis - tej zoo los yog tsis zoo muaj nuj nqis. Nyob rau hauv cov ntaub ntawv ntawm kev nrhiav tsuas yog ib qho hauv paus txhais hais tias cov keeb kwm ntawm tib. Lub orientation ntawm tus nkhaus nyob rau hauv ib tug Cartesian tswj system kuj yuav pre-txiav txim los ntawm lub coefficients ntawm lub W cov hauv paus hniav thiab T. Yog hais tias W muaj ib tug zoo nqi, lub ob ceg ntawm lub parabola yog qhia upwards. Yog hais tias W muaj ib tug tsis zoo nqi, - downwards. Tsis tas li ntawd, yog hais tias tus coefficient B muaj ib tug zoo kos npe rau, nyob rau W kuj zoo, lub kab los sib ntsib ntawm lub parabola muaj nuj nqi yog nyob rau hauv lub "y" los ntawm "-" rau infinity "+" infinity, "c" rau hauv lub chav rho tawm infinity rau pes tsawg. Yog hais tias T - zoo nqi, thiab W - yog tsis zoo, nyob rau lwm yam sab ntawm lub abscissa.