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Kwadraatsvoltooiing

Ons het gesien dat die vergelyking in die vorm:

a 2 x 2 - b 2

bekend is as die verskil in vierkante en kan as volg gefaktoriseer word:

( a x - b ) ( a x + b ) .

Hierdie eenvoudige faktorisering lei na 'n ander tegniek om kwadratiese vergelykings op te los wat bekend staan ​​as kwadraatsvoltooiing .

Ons wys met 'n eenvoudige voorbeeld, deur te probeer om vir x op te los in:

x 2 - 2 x - 1 = 0 .

Ons kan nie maklik faktore van hierdie term vind nie, maar die eerste twee terme lyk soortgelyk aan die eerste twee terme van die volmaakte vierkant:

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

Ons kan egter kul en 'n volmaakte vierkant skep deur 2 aan beide kante van die vergelyking te voeg.

x 2 - 2 x - 1 = 0 x 2 - 2 x - 1 + 2 = 0 + 2 x 2 - 2 x + 1 = 2 ( x - 1 ) 2 = 2 ( x - 1 ) 2 - 2 = 0

Ons weet dat:

2 = ( 2 ) 2

wat beteken dat:

( x - 1 ) 2 - 2

is 'n verskil van vierkante. Ons kan dus skryf:

( x - 1 ) 2 - 2 = [ ( x - 1 ) - 2 ] [ ( x - 1 ) + 2 ] = 0 .

Die oplossing vir x 2 - 2 x - 1 = 0 is dus:

( x - 1 ) - 2 = 0

of

( x - 1 ) + 2 = 0 .

Dit beteken x = 1 + 2 of x = 1 - 2 . Hierdie voorbeeld toon die gebruik van kwadraatsvoltooiing om 'n kwadratiese vergelyking op te los.

Metode: los kwadratiese vergelykings op deur kwadraatsvoltooing

  1. Skryf die vergelyking in die vorm a x 2 + b x + c = 0 . bv. x 2 + 2 x - 3 = 0
  2. Neem die konstante oor na die regterkant van die vergelyking. Bv. x 2 + 2 x = 3
  3. Indien nodig stel die koëffisiënt van x 2 = 1, deur te deel deur die bestaande koëffisiënt.
  4. Neem die helfte van die koëffisiënt van die x -term, kwadreer dit en voeg dit aan beide kante van die vergelyking. Bv. in x 2 + 2 x = 3 , die helfte van die koëffisiënt van die x -term is 1 en 1 2 = 1 . Daarom voeg ons 1 aan albei kante by om x 2 + 2 x + 1 = 3 + 1 te kry.
  5. Skryf die linkerkant as 'n volkome vierkant: ( x + 1 ) 2 - 4 = 0
  6. Jy moet dan in staat wees om die vergelyking in terme van die verskil in vierkante te faktoriseer en dan vir x op te los: ( x + 1 - 2 ) ( x + 1 + 2 ) = 0

Los op:

x 2 - 10 x - 11 = 0

deur kwadraatsvoltooiing.

  1. x 2 - 10 x - 11 = 0
  2. x 2 - 10 x = 11
  3. Die koëffisiënt van die term x 2 is 1.

  4. Die koëffisiënt van die term x is -10. Helfte van die koëffisiënt van die term x sal wees ( - 10 ) 2 = - 5 en die kwadraat sal wees ( - 5 ) 2 = 25 . Dus:

    x 2 - 10 x + 25 = 11 + 25
  5. ( x - 5 ) 2 - 36 = 0
  6. ( x - 5 ) 2 - 36 = 0
    [ ( x - 5 ) + 6 ] [ ( x - 5 ) - 6 ] = 0
  7. [ x + 1 ] [ x - 11 ] = 0 x = - 1 of x = 11

Los op:

2 x 2 - 8 x - 16 = 0

deur kwadraatsvoltooiing.

  1. 2 x 2 - 8 x - 16 = 0
  2. 2 x 2 - 8 x = 16
  3. Die koëffisiënt van die term x 2 is 2. Deel dus beide kante deur 2:

    x 2 - 4 x = 8
  4. Die koëffisiënt van die term x is -4; ( - 4 ) 2 = - 2 en ( - 2 ) 2 = 4 . Dus:

    x 2 - 4 x + 4 = 8 + 4
  5. ( x - 2 ) 2 - 12 = 0
  6. [ ( x - 2 ) + 12 ] [ ( x - 2 ) - 12 ] = 0
  7. [ x - 2 + 12 ] [ x - 2 - 12 ] = 0 x = 2 - 12 or x = 2 + 12
  8. Laat die linkerkant as 'n volkome vierkant geskryf

    ( x - 2 ) 2 = 12
  9. x - 2 = ± 12
  10. Dus x = 2 - 12    of     x = 2 + 12

    Vergelyk met antwoord in stap 7.

Khan academy video on solving quadratics - 1

Kwadraatsvoltooiing oefeninge

Los die volgende vergelykings op deur kwadraatsvoltooiing:

  1. x 2 + 10 x - 2 = 0
  2. x 2 + 4 x + 3 = 0
  3. x 2 + 8 x - 5 = 0
  4. 2 x 2 + 12 x + 4 = 0
  5. x 2 + 5 x + 9 = 0
  6. x 2 + 16 x + 10 = 0
  7. 3 x 2 + 6 x - 2 = 0
  8. z 2 + 8 z - 6 = 0
  9. 2 z 2 - 11 z = 0
  10. 5 + 4 z - z 2 = 0

Questions & Answers

where we get a research paper on Nano chemistry....?
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nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
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Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Preparation and Applications of Nanomaterial for Drug Delivery
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Application of nanotechnology in medicine
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ya I also want to know the raman spectra
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please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
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I think
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Nasa has use it in the 60's, copper as water purification in the moon travel.
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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scanning tunneling microscope
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what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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The nanotechnology is as new science, to scale nanometric
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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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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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in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
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Source:  OpenStax, Siyavula textbooks: wiskunde (graad 11). OpenStax CNX. Sep 20, 2011 Download for free at http://cnx.org/content/col11339/1.4
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