Faktorisering

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Inleiding

In graad 10 het ons gekyk na die oplos van lineêre vergelykings, kwadratiese vergelykings, eksponentieële vergelykings en lineêre ongelykhede. Hierdie hoofstuk bo op daardie werk. Ons kyk na verskillende metodes om kwadratiese vergelykings om te los.

Oplos deur faktorisering

Die oplos van kwadratiese vergelylings deur faktorisering was behandel in Graad 10. Kom ons doen gou 'n voorbeeld om jou geheue te verfris.

Losop die vergelyking, $2 x^{2} - 5 x - 12 = 0$ .

Bepaal of die vergelyking gemene faktore het
Hierdie vergelyking het geen gemene faktore nie.
Bepaal of the vergelyking in die vorm $a x^{2} + b x + c$ met $a > 0$
Die verglyking is in die voorgeskrewe vorm, met $a = 2$ , $b = - 5$ en $c = - 12$ .
Vaktoriseer die kwadrate
$2 x^{2} - 5 x - 12$ het faktore van die vorm:

$(2 x + s) (x + v)$

met $s$ en $v$ konstantes wat bepaal moet word. Dit word vermenigvuldig om

$2 x^{2} + (s + 2 v) x + s v$

te gee. Ons sien dat $s v = - 12$ en $s + 2 v = - 5$ . Hierdie is 'n stel gelyktydige vergelykings in $s$ en $v$ , dit is maklik om numeries op te los. Al die opsies vir $s$ en $v$ word hieronder oorweeg.

$s$ $v$ $s + 2 v$

2 -6 -10

-2 6 10

3 -4 -5

-3 4 5

4 -3 -2

-4 3 2

6 -2 2

-6 2 -2

Ons sien dat die kombinasie van $s = 3$ en $v = - 4$ gee vir ons $s + 2 v = - 5$ .
Skryf die vergelyking met die faktore
$(2 x + 3) (x - 4) = 0$
Los op die vergelyking
Indien twee hakkies vermenigvuldig word en 0 gee, moet een van die hakkies 0 wees, dus

$2 x + 3 = 0$

of

$x - 4 = 0$

Dus, $x = - \frac{3}{2}$ of $x = 4$
Die finale antwoord is
Die oplossing van $2 x^{2} - 5 x - 12 = 0$ is $x = - \frac{3}{2}$ of $x = 4$ .

Dit is belangrik om te onthou dat 'n kwadratiese vergelyking in die vorm $a x^{2} + b x + c = 0$ moet wees voor ons dit kan op los met die metodes.

Los op $a$ : $a (a - 3) = 10$

Herskryf die vergelyking in die vorm $a x^{2} + b x + c = 0$
Verwyder die hakkies en kry al die terme aan een kant van die gelykaanteken.

$a^{2} - 3 a - 10 = 0$
Faktoriseer die drieterm
$(a + 2) (a - 5) = 0$
Los op die vergelyking
$a + 2 = 0$

of

$a - 5 = 0$

Los die twee lineêre vergelykings op en kontroleer die antwoorde in die oorspronklike vergelyking.
Skryf die finale antwoord neer
Dus, $a = - 2$ of $a = 5$

Los op $b$ : $\frac{3 b}{b + 2} + 1 = \frac{4}{b + 1}$

Deel beide kante met die KGV
$\frac{3 b (b + 1) + (b + 2) (b + 1)}{(b + 2) (b + 1)} = \frac{4 (b + 2)}{(b + 2) (b + 1)}$
Bepaal die beperkings
Die noemers is dieselfde, daarom moet die tellers ook dieselfde wees.

Maar $b \neq - 2$ en $b \neq - 1$
Vereenvoudig die vergelyking na die standaard vorm
$\begin{matrix} 3 b^{2} + 3 b + b^{2} + 3 b + 2 & = & 4 b + 8 \\ 4 b^{2} + 2 b - 6 & = & 0 \\ 2 b^{2} + b - 3 & = & 0 \end{matrix}$
Vaktoriseer die drieterm en los op die vergelyking
$\begin{matrix} (2 b + 3) (b - 1) & = & 0 \\ 2 b + 3 = 0 & o f & b - 1 = 0 \\ b = \frac{- 3}{2} & o f & b = 1 \end{matrix}$
Kontroleer die oplossing in die oorspronklike vergelyking
Albei oplossing is geldig

Dus, $b = \frac{- 3}{2}$ of $b = 1$

Oplossing deur faktorisering

Los op die volgende kwadratiese vergelykings op deur faktorisering. Sommige van die antwoorde kan gelos word in die wortel vorm.

$2 y^{2} - 61 = 101$
$2 y^{2} - 10 = 0$
$y^{2} - 4 = 10$
$2 y^{2} - 8 = 28$
$7 y^{2} = 28$
$y^{2} + 28 = 100$
$7 y^{2} + 14 y = 0$
$12 y^{2} + 24 y + 12 = 0$
$16 y^{2} - 400 = 0$
$y^{2} - 5 y + 6 = 0$
$y^{2} + 5 y - 36 = 0$
$y^{2} + 2 y = 8$
$- y^{2} - 11 y - 24 = 0$
$13 y - 42 = y^{2}$
$y^{2} + 9 y + 14 = 0$
$y^{2} - 5 k y + 4 k^{2} = 0$
$y (2 y + 1) = 15$
$\frac{5 y}{y - 2} + \frac{3}{y} + 2 = \frac{- 6}{y^{2} - 2 y}$
$\frac{y - 2}{y + 1} = \frac{2 y + 1}{y - 7}$

Questions & Answers

what is homeostasis?

Samuel Reply

what's physiology

AminchiSunday Reply

what is physiology

AminchiSunday

physically is the study of the function of the body

Najaatu

that is what I want ask

YAU

u are wright

YAU

pls what are the main treatment of hiccups

YAU

physiology is the study of the function of the body

Najaatu

hiccups happen when something irritates the nerves that course your diaphragm to contract

Najaatu

how did hypothalamus manege to control all activities of the various hormones

malual

how can I treat pain a patient feels after eating meals

Namuli Reply

how do I treat a three year old baby of skin infection?

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It depends on the type of infection. Bacterial, fungal, parasitic or viral?

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if you can share the sign ad symptoms of the skin infection then u geh the treatment cox they're different sign ad symptoms of skin infection with different treatment

prostaglandin and fever

Maha Reply

yes

rayyanu

welcome sir

rayyanu

prostaglandin E2 is the final mediator.

Lemlem

prostaglandin E2 is the final mediator of fever.

Lemlem

yes

Agabi

good evening

Jediel

Discuss the differences between taste and flavor, including how other sensory inputs contribute to our perception of flavor.

John Reply

taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.

Faith

While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma

Kamara

which drugs can we use for ulcers

Ummi Reply

omeprazole

Kamara

what

Renee

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is a drug

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of anti-ulcer

Kamara

Omeprazole Cimetidine / Tagament For the complicated once ulcer - kit

Patrick

what is the function of lymphatic system

Nency Reply

Not really sure

Eli

to drain extracellular fluid all over the body.

asegid

The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of

asegid

to transport fluids fats proteins and lymphocytes to the blood stream as lymph

Adama

what is anatomy

Oyindarmola Reply

Anatomy is the identification and description of the structures of living things

Kamara

what's the difference between anatomy and physiology

Oyerinde Reply

Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.

AI-Robot

what is enzymes all about?

Mohammed Reply

Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems

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yes

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how does the stomach protect itself from the damaging effects of HCl

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little girl okay how does the stomach protect itself from the damaging effect of HCL

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it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL

Kamara

function of digestive system

Ali Reply

function of digestive

Ali

the diagram of the lungs

Adaeze Reply

what is the normal body temperature

Diya Reply

37 degrees selcius

Xolo

37°c

Stephanie

please why 37 degree selcius normal temperature

Mark

36.5

Simon

37°c

Iyogho

the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature

Stephanie

37A c

Wulku

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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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$s$	$v$	$s + 2 v$
2	-6	-10
-2	6	10
3	-4	-5
-3	4	5
4	-3	-2
-4	3	2
6	-2	2
-6	2	-2