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Oplos van kwadratiese vergelykings

'n Kwadratiese vergelyking, is 'n vergelyking waar die mag van die veranderlike hoogstens 2 is. Die volgende is voorbeelde van kwadratiese vergelykings.

2 x 2 + 2 x = 1 2 - x 3 x + 1 = 2 x 4 3 x - 6 = 7 x 2 + 2

Kwadratiese vergelykings verskil van lineêre vergelykings daarin dat 'n lineêre vergelyking slegs een oplossing het, terwyl ‘n kwadratiese vergelyking hoogstens 2 oplossings het. Daar is spesiale gevalle waar 'n kwadratiese vergelyking slegs een oplossing het.

Om 'n kwadratiese vergelyking op te los, herskryf ons dit met 'n 0 aan die een kant van die gelykaanteken en die produk van twee lineêre uitdrukkings, in hakies, aan die anderkant. Ons weet byvoorbeeld dat:

( x + 1 ) ( 2 x - 3 ) = 2 x 2 - x - 3

Om op te los:

2 x 2 - x - 3 = 0

moet ons in staat wees om 2 x 2 - x - 3 te herskryf as ( x + 1 ) ( 2 x - 3 ) , en ons weet reeds hoe om dit te doen.

Ondersoek: faktorisering van 'n kwadratiese uitdrukking

Faktoriseer die volgende kwadratiese uitdrukkings:

  1. x + x 2
  2. x 2 + 1 + 2 x
  3. x 2 - 4 x + 5
  4. 16 x 2 - 9
  5. 4 x 2 + 4 x + 1

As jy 'n kwadratiese uitdrukking kan faktoriseer, is jy een stap weg daarvan om 'n kwadratiese vergelyking op te los. Byvoorbeeld, x 2 - 3 x + 2 = 0 kan geskryf word as ( x - 1 ) ( x - 2 ) = 0 . Dit beteken dat x - 1 = 0 of x - 2 = 0 , wat x = 1 en x = 2 gee as die 2 oplossings van die kwadratiese vergelyking x 2 - 3 x + 2 = 0 .

Metode: oplos van kwadratiese vergelykings

  1. Deel heel eerste die hele vergelyking deur enige gemene faktore van die koëffisiënte, ten einde 'n vergelyking te kry van die vorm a x 2 + b x + c = 0 waar a , b en c geen gemeenskaplike faktore het nie. Byvoorbeeld, 2 x 2 + 4 x + 2 = 0 kan geskryf word as x 2 + 2 x + 1 = 0 deur te deel met 2.
  2. Skryf a x 2 + b x + c in terme van sy faktore ( r x + s ) ( u x + v ) . Dit beteken ( r x + s ) ( u x + v ) = 0 .
  3. Wanneer ons die vergelyking geskryf het in die vorm ( r x + s ) ( u x + v ) = 0 , volg dit dat die oplossing sal wees x = - s r of x = - v u .
  4. Vervang elke moontlike waarde van die oplossing in die oorspronklike vergelyking in om te toets of dit 'n geldige oplossing is.

Oplossing (wortels) van kwadratiese vergelykings

'n Kwadratiese vergelyking het 2 wortels omdat enige een van die 2 waardes die vergelyking kan bevredig.

Khan akademie video oor vergelykings - 3

Los op vir x : 3 x 2 + 2 x - 1 = 0 .

  1. Ons het gesien die faktore van 3 x 2 + 2 x - 1 is ( x + 1 ) and ( 3 x - 1 ) .

  2. ( x + 1 ) ( 3 x - 1 ) = 0
  3. Ons het

    x + 1 = 0

    of

    3 x - 1 = 0

    Dus, x = - 1 of x = 1 3 .

  4. As ons die antwoorde instel in die oorspronklike vergelyking in, vind ons die vergelyking is waar vir beide antwoorde.
  5. 3 x 2 + 2 x - 1 = 0 vir x = - 1 of x = 1 3 .

Dit mag gebeur dat die vergelyking met die eerste oogopslag nie soos 'n kwadratiese vergelyking lyk nie, maar deur 'n paar bewerkings in een verander kan word. Onthou dat dieselfde bewerking aan elke kant gedoen moet word om die vergelyking geldig (waar) te hou.

Dit mag nodig wees om een of meer van die volgende te doen:

  • Byvoorbeeld,
    a x + b = c x x ( a x + b ) = x ( c x ) a x 2 + b x = c
  • Dit beteken om beide kante te verhef tot die mag - 1 . Byvoorbeeld,
    1 a x 2 + b x = c ( 1 a x 2 + b x ) - 1 = ( c ) - 1 a x 2 + b x 1 = 1 c a x 2 + b x = 1 c
  • Dit beteken om weerskante te verhef tot die mag 2. Byvoorbeeld,
    a x 2 + b x = c ( a x 2 + b x ) 2 = c 2 a x 2 + b x = c 2

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
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yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
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are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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Do you know which machine is used to that process?
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for screen printed electrodes ?
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of graphene you mean?
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or in general
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in general
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Graphene has a hexagonal structure
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Source:  OpenStax, Siyavula textbooks: wiskunde (graad 10) [caps]. OpenStax CNX. Aug 04, 2011 Download for free at http://cnx.org/content/col11328/1.4
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