Home MATHEMATICS TOPIC 3: QUADRATIC EQUATIONS ~ MATHEMATICS FORM 2

TOPIC 3: QUADRATIC EQUATIONS ~ MATHEMATICS FORM 2

25/09/2020

3852

TOPIC 3: QUADRATIC EQUATIONS ~ MATHEMATICS FORM 2

Solving quadratic equations can be difficult, but luckily there are several different methods that we can use depending on what type of quadratic that we are trying to solve.

The four methods of solving a quadratic equation are factoring, using the square roots, completing the square and the quadratic formula.

Solving Equations

The standard form of a Quadratic equation is ax² + bx + c =0whereby a, b, c are known values and ‘a’ can’t be 0. ‘x’ is a variable (we don’t know it yet). a is the coefficient of x² , b is the coefficient of x and c is a constant term. Quadratic equation is also called an equation of degree 2 (because of the 2 on x). There are several methods which are used to find the value of x. These methods are:

by Factorization
by completing the square
by using quadratic formula

The Solution of a Quadratic Equation by Factorization

Determine the solution of a quadratic equation by factorization

We can use any of the methods of factorization we learnt in previous chapter. But for simplest we will factorize by splitting the middle term. For Example: solve for x, x² + 4x = 0

solution Since the constant term is 0 we can take out x as a common factor.

So, x² + 4x = x(x + 4) = 0. This means the product of x and (x + 4) is 0. Then, either x = 0 or x + 4 = 0. If x + 4 = 0 that is x = -4. Therefore the solution is x = 0 or x = -4.

Example 1

Solve the equation: 3x² =- 6x – 3.

first rearrange the equation in its usual form.

that is:

3x² = – 6x – 3

3x² + 6x + 3 = 0

now, factorize the equation by splitting the middle term. Let us find two numbers whose product is 9

and their sum is 6. The numbers are 3 and 3. Hence the equation 3x² + 6x + 3 = 0 can be written as:

3x² + 3x + 3x + 3 = 0

3x(x + 1) + 3(x + 1) = 0

(3x + 3)(x + 1) (take out common factor which is (x + 1))

either (3x + 3) = 0 or (x + 1) = 0

therefore 3x = -3 or x = -1

x = -1 (divide by 3 both sides) or x = -1

Therefore, since the values of x are identical then x = -1.

Example 2

solve the equation 10 – 3y – 1 = 0 by factorization.

Solution

Two numbers whose product is -10 and their sum is -3 are 2 and -5.

Then, we can write the equation 10y² – 3y – 1 = 0 as:

2y(5y + 1) – 1(5y + 1) = 0

(2y – 1)(5y + 1) = 0

ORDINARY LEVEL FULL NOTES ALL SUBJECTS (FORM 1 - 4)

Therefore, either 2y – 1 = 0 or 5y + 1 = 0

Example 3

solve the following quadratic equation by factorization: 4x² – 20x + 25 = 0.

Solution

We need to split the middle term by the two numbers whose product is 100 and their sum is -20. The numbers are -10 and -10.

The equation can be written as:

4x² -10x – 10x + 25 = 0

2x(2x – 5) – 5(2x -5) = 0

(2x – 5)(2x – 5) (take out common factor. The resulting factors are identical. This is a perfect square)

since it is a perfect square, then we take one factor and equate it to 0. That is:

2x – 5 = 0

2x = 5 then, divide by 2 both sides.

Therefore

Example 4

solve the equation x² – 16 = 0.

Solution

We can write the equation as x² – 4² = 0. This is a difference of two squares. The difference of two squares is an identity of the form:

a² – b² = (a – b)(a + b).

So, x² – 4² = (x – 4)(x + 4) = 0

Now, either x– 4 = 0 or x + 4 = 0

Therefore x = 4 or x = -4

The Solution of a Quadratic Equation by Completing the Square

Find the solution of a quadratic equation by completing the square

Completing the square.

Example 5

Add a term that will make the following expression a perfect square: x² – 8x

find a term that must be added to make the following expression a perfect square: x² + 10x

Example 6

solve the following quadratic equation by completing the square: x² + 4x + 1 = 0

Example 7

solve by completing the square: 3x² + 7x – 6 = 0

General Solution of Quadratic Equations

The Quadratic Formula

Derive the quadratic formula

The special quadratic formula used for solving quadratic equation is:

Quadratic Equations using Quadratic Formula

TOPIC 8: ACCOUNTS ~ MATHEMATICS FORM 3

Solve quadratic equations using quadratic formula

Example 8

solve 5x² – 8x + 3 = 0 by using quadratic formula.

Example 9

solve this quadratic equation by using quadratic formula: 3x² = – 7x – 4

Word problems leading to quadratic equations

Given a word problem; the following steps are to be used to recognize the type of equation.

Step1: choose the variables to represent the information

Step 2: formulate the equation according to the information given

Step 3: solve the equation by using any of the method you know

In order to be sure with your answers, check if the solution you obtained is correct.

Example 10

the length of a rectangular plot is 8 centimeters more than the width. If the area of a plot is 240cm², find the dimensions of length and width.

Solution

Let the width be x

The length of a plot is 8 more than the width, so the length of a plot be x + 8

We are given the area of a plot = 240cm² and the area of a rectangle is given by length ×width

then (x + 8) × x = 240

x² + 8x = 240

rearrange the equation

x² + 8x – 240 = 0

then solve the equation to find the value of x

Solving by splitting the middle term, two numbers whose product is -240 and their sum is 8, the number

are -12 and 20

our equation becomes; x² + 20x – 12x – 240 = 0

x(x + 20) – 12(x + 20) = 0

either (x – 12) = 0 or (x + 20) = 0

x = 12 or x = -20

since we don’t have negative dimensions, then the width is 12cm and the length is 12 + 8 = 20cm

ALL TIE BOOKS FOR PRIMARY AND SECONDARY SCHOOLS

Therefore the rectangular plot has the length of 20cm and the width of 12cm.

Example 11

A
piece of wire 40cm long is cut into two parts and each part is then
bent into a square. If the sum of the areas of these squares is 68
square centimeters, find the lengths of the two pieces of wire.

Exercise 1

1. Solve each of the following quadratic equations by using factorization method:

-6x²+ 23x – 20 = 0
X²– x -12 = 0

2. Solve these equations by completing the square:

YOU MUST ALSO READ THIS

TOPIC 1: COORDINATE GEOMETRY ~ MATHEMATICS FORM 4

TOPIC 1: COORDINATE GEOMETRY ~ MATHEMATICS FORM 4 Equation of a ...

TOPIC 4: LOGARITHMS ~ MATHEMATICS FORM 2

TOPIC 4: LOGARITHMS ~ MATHEMATICS FORM 2 We always ask ourselves, ...

MATHEMATICS NOTES FOR ORDINARY LEVEL (FORM 1 – 4)

MATHEMATICS NOTES FOR ORDINARY LEVEL (FORM 1 - 4) MATHEMATICS NOTES ...

480 COMMENTS

Lajki na Instagram 14/10/2022 at 17:21

This is my first time pay a visit at here and i am truly happy to read all at one place.

Reply
Server Associate Question 13/10/2022 at 15:37

ivermectin for canaries ivermectin for sale

Reply
https://forex-blog-iran.com/ 11/10/2022 at 20:31

Having read this I thought it was really informative. I appreciate you taking the time and effort to put this content together. I once again find myself spending way too much time both reading and commenting. But so what, it was still worthwhile!

Reply
bo dam motorola 11/10/2022 at 11:09

Hello there! This is my first visit to your blog! We are a collection of volunteers and starting a new initiative in a community in the same niche.Your blog provided us beneficial information to work on. Youhave done a marvellous job!

Reply
removals Nottinghamshire 11/10/2022 at 00:01

online pharmacy busted pharmacy on line canada

Reply
van removals 10/10/2022 at 17:50

I enjoy what you guys are usually up too. This type of clever work and exposure!Keep up the superb works guys I’ve incorporated you guys to my blogroll.

Reply
visit website 10/10/2022 at 16:25

Useful material. Thank you!essay writing service canada essaytyper write my thesis

Reply
california calibration 10/10/2022 at 15:49

Olá! Publiquei um trecho de 24 segundos para divulgar o novo clipe da Juliette e ele foi bloqueado. Vocês me ajudam? Vocês me autorizam? Foi no meus Reels no @popluvbrasil no Instagram.

Reply
patriotic shirts kids 10/10/2022 at 09:00

Normally I do not learn post on blogs, however I wouldlike to say that this write-up very compelled me to try and doit! Your writing taste has been amazed me. Thank you, quite great post.

Reply
V&D Removals 08/10/2022 at 00:27

Thanks for sharing your info. I truly appreciate your efforts and I am waiting for your further post thanks once again.

Reply
visit website 06/10/2022 at 22:05

Heya i抦 for the first time here. I found this board and I find It really useful & it helped me out much. I hope to give something back and aid others like you aided me.

Reply
lap dat camera 06/10/2022 at 16:58

clomiphene tablets clomid for sale clomiphene online

Reply
con ke thep san 06/10/2022 at 11:49

Ouecln – poems to write essays on Cbwcum babbbg

Reply
click here 06/10/2022 at 07:45

Thanks for the marvelous posting! I really enjoyed reading it, you are a great author.I will be sure to bookmark your blog and may come back in the foreseeable future.I want to encourage you continue your great posts,have a nice morning!

Reply
Giai phap chong rung toc 06/10/2022 at 02:26

Generally I don’t read article on blogs, but I wish to say that this write-up very compelled me to check out and do it! Your writing style has been amazed me. Thank you, very nice article.

Reply
cham soc da chuan y khoa 05/10/2022 at 21:12

whoah this blog is fantastic i really like reading your posts. Stay up the good work! You know, many persons are searching around for this info, you can aid them greatly.

Reply
mang chong tham hdpe 05/10/2022 at 16:03

Hi there friends, how is the whole thing, and whatyou want to say about this article, in my view its genuinely awesome designed for me.

Reply
software para control de activos fijos gratis 05/10/2022 at 14:22

apartments that allow pets near me rentberry scam ico 30m$ raised rollingwood apartments

Reply
removals uk 05/10/2022 at 04:15

Herkes bol takipçi ister, size en yüksek miktarlarda takipçi atacak ve en cüzi fiyattan instagram takipçi satın almanıza yarayacak panelimiz hızlıdır.

Reply
removals Nottinghamshire 04/10/2022 at 21:55

Thanks for the auspicious writeup. It actually used to be a leisure account it.Look complex to far brought agreeable from you! By the way,how can we keep in touch?

Reply
VD Removals 04/10/2022 at 15:34

I am not certain where you’re getting your info, but great topic. I needs to spend some time learning more or figuring out more. Thanks for magnificent information I was looking for this information for my mission.

Reply
vavada kazino oficial`nyj sajt vhod 04/10/2022 at 13:38

great points altogether, you just gained a brand new reader. What would you recommend about your post that you made some days ago? Any positive?

Reply
Percy Young 04/10/2022 at 07:05

On the other hand, betting on high college athletic eventsremains illegal.

Reply
Tax on Foreign income 04/10/2022 at 02:29

Attractive section of content. I just stumbled upon your blog and in accession capital to assert that I acquirein fact enjoyed account your blog posts. Anyway I will be subscribing to your feeds andeven I achievement you access consistently quickly.

Reply
Self Assessment tax Return Accountant 03/10/2022 at 20:07

nolvadex only pct nolvadex for sale – what happens when you stop taking tamoxifen

Reply
Cheap cbd vape disposables 02/10/2022 at 23:59

Ahaa, its good conversation concerning this piece of writing here at thisblog, I have read all that, so at this time me also commenting at this place.

Reply
potting sheds 02/10/2022 at 15:55

It’s difficult to find experienced people on this subject, but you sound like you know what you’re talking about! Thanks

Reply
cong tac thong minh 02/10/2022 at 13:02

apartments for rent in green bay rentberry scam ico 30m$ raised apartments in sherwood ar

Reply
Cua go cong nghiep 02/10/2022 at 07:44

ms 400 headphones ik niet gezien lydstyrkenivГҐet er moderat

Reply
cong ty quay phim da nang 02/10/2022 at 02:13

that I have really enjoyed browsing your blog posts.

Reply