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'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 2 x 2 + 2 x = 1 2 - x 3 x + 1 = 2 x 4 3 x - 6 = 7 x 2 + 2
(1)

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

Om op te los:

2 x 2 - x - 3 = 0 2 x 2 - x - 3 = 0
(3)

moet ons in staat wees om 2x2-x-32x2-x-3 te herskryf as (x+1)(2x-3)(x+1)(2x-3), en ons weet reeds hoe om dit te doen.

### Ondersoek: Faktorisering van 'n Kwadratiese Uitdrukking

1. x + x 2 x + x 2
2. x 2 + 1 + 2 x x 2 + 1 + 2 x
3. x 2 - 4 x + 5 x 2 - 4 x + 5
4. 16 x 2 - 9 16 x 2 - 9
5. 4 x 2 + 4 x + 1 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, x2-3x+2=0x2-3x+2=0 kan geskryf word as (x-1)(x-2)=0(x-1)(x-2)=0. Dit beteken dat x-1=0x-1=0 of x-2=0x-2=0, wat x=1x=1 en x=2x=2 gee as die 2 oplossings van die kwadratiese vergelyking x2-3x+2=0x2-3x+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 ax2+bx+c=0ax2+bx+c=0 waar aa, bb en cc geen gemeenskaplike faktore het nie. Byvoorbeeld, 2x2+4x+2=02x2+4x+2=0

kan geskryf word as x2+2x+1=0x2+2x+1=0

deur te deel met 2.
2. Skryf ax2+bx+cax2+bx+c in terme van sy faktore (rx+s)(ux+v)(rx+s)(ux+v). Dit beteken (rx+s)(ux+v)=0(rx+s)(ux+v)=0.
3. Wanneer ons die vergelyking geskryf het in die vorm (rx+s)(ux+v)=0(rx+s)(ux+v)=0, volg dit dat die oplossing sal wees x=-srx=-sr of x=-vux=-vu.
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.

Figure 1
Khan Akademie video oor vergelykings - 3

#### Exercise 1: Oplos van Kwadratiese Vergelykings

Los op vir xx: 3x2+2x-1=03x2+2x-1=0.

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:

• Vermenigvuldig weerskante: Byvoorbeeld,
ax+b=cxx(ax+b)=x(cx)ax2+bx=cax+b=cxx(ax+b)=x(cx)ax2+bx=c
(7)
• Kry weerskante die resiproke: Dit beteken om beide kante te verhef tot die mag -1-1. Byvoorbeeld,
1ax2+bx=c(1ax2+bx)-1=(c)-1ax2+bx1=1cax2+bx=1c1ax2+bx=c(1ax2+bx)-1=(c)-1ax2+bx1=1cax2+bx=1c
(8)
• Kwadreer weerskante: Dit beteken om weerskante te verhef tot die mag 2. Byvoorbeeld,
ax2+bx=c(ax2+bx)2=c2ax2+bx=c2ax2+bx=c(ax2+bx)2=c2ax2+bx=c2
(9)

Hierdie strategieë kan op verskillende wyses gekombineer word en die sekerste manier om jou intuïsie te ontwikkel oor wat die beste ding is om te doen, is om te oefen. 'n Gekombineerde stel bewerkings kan byvoorbeeld wees:

1 a x 2 + b x = c ( 1 a x 2 + b x ) - 1 = ( c ) - 1 ( kry weerskante die resiprook ) a x 2 + b x 1 = 1 c a x 2 + b x = 1 c ( a x 2 + b x ) 2 = ( 1 c ) 2 ( kwadreer weerskante ) a x 2 + b x = 1 c 2 1 a x 2 + b x = c ( 1 a x 2 + b x ) - 1 = ( c ) - 1 ( kry weerskante die resiprook ) a x 2 + b x 1 = 1 c a x 2 + b x = 1 c ( a x 2 + b x ) 2 = ( 1 c ) 2 ( kwadreer weerskante ) a x 2 + b x = 1 c 2
(10)

#### Exercise 2: Oplos van Kwadratiese Vergelykings

Los op vir xx: x+2=xx+2=x.

#### Exercise 3: Oplos van Kwadratiese Vergelykings

Los die vergelyking op: x2+3x-4=0x2+3x-4=0.

#### Exercise 4: Oplos van Kwadratiese Vergelykings

Vind die wortels van die kwadratiese vergelyking 0=-2x2+4x-20=-2x2+4x-2.

1. Los op vir xx: (3x+2)(3x-4)=0(3x+2)(3x-4)=0

Kliek hier vir die oplossing
2. Los op vir xx: (5x-9)(x+6)=0(5x-9)(x+6)=0

Kliek hier vir die oplossing
3. Los op vir xx: (2x+3)(2x-3)=0(2x+3)(2x-3)=0

Kliek hier vir die oplossing
4. Los op vir xx: (2x+1)(2x-9)=0(2x+1)(2x-9)=0

Kliek hier vir die oplossing
5. Los op vir xx: (2x-3)(2x-3)=0(2x-3)(2x-3)=0

Kliek hier vir die oplossing
6. Los op vir xx: 20x+25x2=020x+25x2=0

Kliek hier vir die oplossing
7. Los op vir xx: 4x2-17x-77=04x2-17x-77=0

Kliek hier vir die oplossing
8. Los op vir xx: 2x2-5x-12=02x2-5x-12=0

Kliek hier vir die oplossing
9. Los op vir xx: -75x2+290x-240=0-75x2+290x-240=0

Kliek hier vir die oplossing
10. Los op vir xx: 2x=13x2-3x+14232x=13x2-3x+1423

Kliek hier vir die oplossing
11. Los op vir xx: x2-4x=-4x2-4x=-4

Kliek hier vir die oplossing
12. Los op vir xx: -x2+4x-6=4x2-5x+3-x2+4x-6=4x2-5x+3

Kliek hier vir die oplossing
13. Los op vir xx: x2=3xx2=3x

Kliek hier vir die oplossing
14. Los op vir xx: 3x2+10x-25=03x2+10x-25=0

Kliek hier vir die oplossing
15. Los op vir xx: x2-x+3x2-x+3


Kliek hier vir die oplossing
16. Los op vir xx: x2-4x+4=0x2-4x+4=0

Kliek hier vir die oplossing
17. Los op vir xx: x2-6x=7x2-6x=7

Kliek hier vir die oplossing
18. Los op vir xx: 14x2+5x=614x2+5x=6

Kliek hier vir die oplossing
19. Los op vir xx: 2x2-2x=122x2-2x=12

Kliek hier vir die oplossing
20. Los op vir xx: 3x2+2y-6=x2-x+23x2+2y-6=x2-x+2

Kliek hier vir die oplossing

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