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 inflation

Bright Reply

a general and ongoing rise in the level of prices in an economy

AI-Robot

What are the factors that affect demand for a commodity

Florence Reply

price

Kenu

differentiate between demand and supply giving examples

Lambiv Reply

differentiated between demand and supply using examples

Lambiv

what is labour ?

Lambiv

how will I do?

Venny Reply

how is the graph works?I don't fully understand

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information

Eliyee

devaluation

Eliyee

WARKISA

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Lambiv

multiple choice question

Aster Reply

appreciation

Eliyee

explain perfect market

Lindiwe Reply

In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,

Ezea

What is ceteris paribus?

Shukri Reply

other things being equal

AI-Robot

When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab

Kelo

Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)

Kelo

yes,thank you

Shukri

Can I ask you other question?

Shukri

what is monopoly mean?

Habtamu Reply

What is different between quantity demand and demand?

Shukri Reply

Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded

Ezea

Shukri

how do you save a country economic situation when it's falling apart

Lilia Reply

what is the difference between economic growth and development

Fiker Reply

Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.

Shukri

production function means

Jabir

What do you think is more important to focus on when considering inequality ?

Abdisa Reply

any question about economics?

Awais Reply

sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏

Asui

it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs

Awais

thank you so much 👍 sir

Asui

In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p

Cornelius

Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .

Feyisa Reply

Answer

Feyisa

Jabir

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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