smArt

sMart We proudly present the Math work within our European project smArt. Students taught Maths and History to their peer students.

Maths Lesson:​ Geometry Names:​ Lisa Bruggeman, Kirsten Van Remoortere, Phebe Stuer, Iris Rooms Topic:​ sinus, cosine rules Material:

PYTHAGORAS rule & geometry I’m going to tell you more about the theory of Pythagoras. This rule only counts in a triangle which is rectangular. The first thing we do is take our triangle and give letters on the sides.

So you you have a b and c. And Pythagoras said that there’s 1 rule: ​It states that the square of the ​hypotenuse​ (the side opposite the ​right angle​) is equal to the sum of the squares of the ​other two sides​.

So then you become A^2 ( to the second )= is equal to B2 + C2 Or when you change the letters you can have B2= A2+C2 Here’s a video of this rule.

There’s also the trigonometry, ​ ​it’s ​a branch of mathematics that also works with triangles and most of the times in a cirkel. There’s the sine that is the length on the y axis. The cosine is the length on the x axis.. There’s also the tangent which you can find when you draw a line when X = 1 and the cotangent can you calculate on the line where Y is equal to 1.Here’s a list of all the relationships in the trigonometry between the corner and the sides.

Trigonometry isn’t always in a circle. You can use sine, cosine and tangent in a triangle but only in a rectangular triangle and always relative an angle. So there are 3 rules and in belgium there’s a simple way to remember , it sounds like SOS CASTOA. In English they call it … and you can see what it stands for.

SOHCAHTOA in het Engels ●

SOH: Sine = Opposite ÷ Hypotenuse and so on

● ●

CAH: Cosine= Adjacent ÷ Hypotenuse TOA: Tangent = Opposite ÷ Adjacent

The adjacent is the length of the side from the unknown angle to the right angle.

The hypotenuse I already said it’s the longest side opposite the right angle.

And the opposite, I think that’s clear? It’s just opposite the unknown angle

So for example, there’s a triangle ABC with B is 90 degrees. You start from alpha so sine= BC on AC, the cosine is AB on AC and the tangent is equal to BC / AB. It isn’t difficult when you know this 3 rules by heart.

I will do a little exercise: there’s a rectangular triangle with an angle of 30 degrees. They ask for the sine of the angle C. You already know that sine is opposite on hypotenuse so that’s AB on AC. Then you become 5 / 10 and that’s equal to 0,5. And that’s it.

Good to know… there’s also an important list to study of relationships between the triangle in the trigonometry. But you can use the rules that we have seen to find the solution.

COSINUS REGEL Theory: Cosine rule In a random triangle which in this case is called ABC, we show the lengths of the sides against the angles A, B en C by a, b & c. We call the angles by A, B & C alpha, beta and gamma. In those random triangles applies: a to second equals b to the second plus c to the second minus 2bc cosine alpha and b to the second = a to the second plus c to the second minus 2ac cosine beta, and also c to the second = a to the second plus b to the second - 2ab cosine gamma.

These are the formulas that we call the cosine formula.

A little demonstration: In the first case alpha is an acute angle. First we draw an altitude line from C. From this point the foot perpendicular to C on AB is D. In this rectangular triangle BCD applies: a² equals |CD|² + |DB|², which equals |CD|² + length (c - |AD|)². |CD| and |AC| we can find in triangle ADC. Meanwhile sinus alpha equals |CD| divided by b while cosine alpha equals |AD| divided by b, which means that |CD| equals b times sinus alpha while |AD| equals b times cosine alpha. Seeing that explains why we suddenly hop to that this equals (b times sinus alpha)² + (c - b times cosine alpha)². This then equals b² sinus² + c² - 2bc cosine alpha + b² + cosine² alpha. Going from there, we can say that b² times (sinus² alpha + cosine² alpha) + c² - 2bc cosine alpha. Seeing as sinus² alpha + cosine² alpha = 1, we can say that b² + c² - 2bc cosine alpha equals a²​.

In the second case alpha is an obtuse angle. With that we also draw an altitude line from C which will cut AB in D. In the rectangular triangle BCD we can find: a² equals |CD|² + |BC|² which then equals |CD|² + (c + |AD|)² because line (c + AD)² equals |BC|². Seeing as we can find |CD| and |AD| we can find in triangle ADC. Sinus (180° - alpha) equals |CD| / b and cosine (180° - alpha) = |AD| / b. That means |CD| equals b times sinus (180° - alpha) and |AD| equals b times cosine (180° - alpha), because they are supplementary angles which are angles that add up to 180°. Knowing all that information, we can say that a² equals (b times sinus alpha)² + (c + (-b times cosine alpha)². And if you would work this out, you can say that a² equals b² + c² - 2bc cosine alpha.

In the last case alpha is a right angle, where we can just solve this with the ​Pythagorean theorem, which was explained before. Just as a reminder, the Pythagorean theorem is: a² equals b² + c². Seeing as cos alpha equals 90 equals 0, we can rewrite this as: a² equals b² + c² - 2bc cos alpha.

We can see the cosine rule as a generalization of the Pythagorean theorem, seeing as the cosine rule also applies for random triangles. You could also say that the Pythagorean theorem is an unique case of the cosine rule, seeing as the Pythagorean only is valid for rectangular triangles.

So in the end, we can say that out of all three cases we have seen that in random triangles a² = b² + c² - 2bc cos alpha is what is proven to work. I know that it is a bit hard, but eventually you will get it.

Exercises: James wants to fly from the airport in Moerbeke (M) to point A with his airplane. Point A and M are 100 km away from each other.

He flies at a speed of 80 km/h. After 1 hour he should be at 20 km away from A. due to the westerly wind, he changed course of 15°.

Due to that he is now in point B, that is more than 20 km away from point A. how far is he away from A now If ABM was a ​right-angled triangle, than we could find x which is the same as the length of AB with Pythagoras. But there is no right-angled triangle.

You can draw a line out of B, now you can split up ΔABM in two right-angled triangle

Measure the unknown length x which is the same as the length of AB. Measure the missing sides in triangle ABC with

Measur the angles from triangle ABC at 1’’ ( significant numbers) accurite if a=6 , b=7 and c=8.

The Wiericker Schans is a former walled army camp from the 17th century in The Netherlands. To determine the with of the building, in between the points P and Q, Jonas goes as followed.

Since he can’t directly see the poles P and Q from A, he adds two more poles B and C, in order that B and C, P and Q lie on a line. The measurements in meter, in between the poles and the angle BÂC you can read on the figure.

Measure PQ

Correction of the exercises:

The history of Ghent

By Nina, Jolien, Felix and Bernd

Ghent -East flanders province -Rivers scheldt and leie -candy’s -old buildings

The castle of the counts -

Filips 1180 vikings building work prison forture

Life in the castle -Little village - RESIDENTS: Lord, maids, servants, groomers,cooks, baker, tower guards, soldiers, hunters, shoe-and saddle makers, hairdresser = mender, chaplain - ⇒ all these residents = rich nobleman -Boring: → stand up early (4 o’clock) sunshine= crucial to getting things done

Spending time in a castle -HUNT: more meat -Workouts from then: jousting game -Lord: -Talks about business with his men -Work within the castle was done -The wife of the lord: -could handle a horse -hunt -lorde was gone= she in charge

Parties in the midd le ages -Organisation: lower classes , a lot of work

-Table: friends or at table → level of importance -Lord and Lady: at the head of wooden table Food: meat, spices, salted fish, beer and wine Clothes: finest fabrics, colourful, jewels, gold/silver ⇒ fashionable and expensive → impress each other -Music: middle age music

It was not like a fairy tail -castle → smelled awful → no privacy → dark/cold → prisoners and torture

Torture inside the castle -Authentic torture vessels (buques de tortura)

Reasons ?

-to make someone confess.

-steal something.

-have lied about something important.

The copper bull (el toro de cobre)

The Guillotine -

Popular Head → middle of the frame Blade dropped Instant, painless The most human method

The Rack -

Most painful wooden frame 2 robes → bottom other 2 → handle on top bound and placed torturer → turn the handle Limbs → dislocated

Video ●

https://youtu.be/GmnSgkVoaRw

Thales

Biography ● Thales was a philosopher and a mathematician ● He was born in Mileto ( Greece ), in 624 BC ● He died in 546 BC

Biography ● Thales was known for having his head in the clouds, the stereotypical “absent-minded professor” ● He introduced geometry in Greece

Milesian School â—? He founded the Milesian School, a school of thought â—? He had a well-known disciple, Pythagoras

What is a similar triangle?

Two triangles are similar if their angles are equal one by one.

They can be of dierent sizes.

Which of these triangles are similar?

Thales’s Theorem

Example

Application to an exercise with shadows The shadow of the objects used form similar triangles.

Example: x

1,5 m 262 m 2,87 m

Application to an exercise with shadows Example:

•C/B=D/A ; 262/2,87=x/1,5 •x=(262⋅1,5)/2,87 ; x=137m

x 1,5 m 2,87 m

262 m

Thales Theorem with an obstacle Now, we are going to understand all this theory due to an easy example.

1. As you can observe, this picture shows two triangles, the blue one and the purple one. They are similar so we can use the Thales Theorem. Does anybody know why the triangles are similar?

2. Well, we need to know which is the building height and to guess it, the exercise gives you some measurements. Does anybody know what we have to do?

SOLUTION As the triangles are similar we need to use the Thales Theorem . What we have to do is compare their sides.

So the building measures 225 metres.

THE NAZARI KINGDOM OF

GRANADA

THE BEGINNING OF THE NAZARI KINGDOM In the Middle Age, the muslims were reigning in Spain but then the christians conquered much of the territory and reduced the muslim kingdom.

The last muslims in Spain were the nazari kingdom of Granada.It was in the south of Spain with its capital in Granada.

The nazari kingdom had 20 sultans in total but it was founded in 1238 by the nasrid noble Muhammad I, called “The Red� who built the amazing palace of La Alhambra.

THEART ARTIN INTHE THENAZARĂ?

NAZARĂ?KINGDOM KINGDOM THE During this period, alcazabas, towers, walls and tiles used to decorate the kingdom.The Islamic art settled in Al-andalus us an wealth.

During this period, alcazabas, towers, walls and letting tiles used to amazing decoratecultural the kingdom.The Islamic art settled in Al-andalus letting us an amazing cultural wealth.

CHARACTERISTICS;

--Sobriety in the outdoor decoration and overloaded decoration in the inside

--The use of new kinds of colums furnished with atauriques

--The use of poor materials

--Every expresiรณn of art is very related to the the islam

Nowadays, we still can see traces of nazarí’s art in our modern streets, houses and art. One easy example that all of you can see are the tiles in our school walls that remind us to the típical nazarí style

Some examples of buildings that still persist are ;

The Målaga’ s Alcazaba

The Turkish baths in Ronda

The Alhambra

ALHAMBRA The History

The beginning of the Alhambra The Alhambra was palace,citadel and fortress, residence of Nazaries Sultans and senior officials.

Reaches its splendor in the second half of the 14th century.

The Lion’s Courtyard

The lion’s courtyard is one of the most characteristics structures in the Alhambra.

Each one of the lions symbolizes one of the months of the year.

There are 12 lions in total.

GRANADA: THE FALLEN KINGDOM The Reconquest ended with the fall of Granada in 1492, wich was ruled by Boabdil, the last nazari sultan.

BOABDIL

ISABEL I

FERNANDO II

THE CAPITULATIONS

PROHIBITIONS AND RULES

MUSLIMS CANNOT HAVE GUNPOWDER

CHRISTIAN HOSTAGES WILL BE FREE

PERMISSIONS AND RIGHTS

MUSLIMS WILL BE JUDGED BY THEIR LAWS

MUSLIMS CAN EXERCISE THEIR RELIGION

Finally, Boabdil gave the keys of the city to the Catholics Monarchs.

In Marbella, on Malaga Street, Muslims gave the keys of the city to Christians in 1485.

“Cry like a woman for what you could not defend like a man�- Aixa.

This is a Spanish proverb commonly used in our country.

SOCIETY

Hj

There were three cultures living in peace: christians, muslims and jews.

Different cultures: Jews' holy day is saturday Whereas for christians celebrated on sundays.

Also muslims diet was peculiar, they had some restrictions about eating pork.

They could cohabit but they had to pay taxes for that

THE END

GEOMETRIC TRANSFORMATION

• A geometric transformation is a one-to-one function whose domain and range are sets of points on the plane

ISOMETRY • An isometry is a geometric transformation that is distant-preserving

TRANSLATION • A translation maps each point of the plane to its image along a given vector called the translation vector

ROTATION • A rotation is an isometry that has a given center 0 and a given angle of rotation α.

• Each point of the plane is mapped onto a point that is rotated around 0 by the angle α

POINT REFLECTION • A point reflection in 0 is an isometry that maps each point of the plane P to a point P’ such that PO ≅ OP’ and P’ lies on the line through PO

REFLECTION • A reflection across a line r of the plane is an isometry that maps each point P of the plane onto a point P’ such that PP’ is perpendicular to r and the distance from P to r is the same as the distance from r to P’

S. FRANCIS’S WAY AND S. JAMES’ WAY

SAINT FRANCIS

◦ Saint Francis’s life ◦ The Franciscan Order

Saint Francis’s Way Discover the places he passed through, where he stopped and then went 350 km, or slightly more, from La Verna and the wooded hills of the Tuscany border to the lovely wide valley of Rieti in Lazio, through the most meaningful sites in the life of St Francis, up the hills and down the valleys of wonderful Umbria, the geographical heart of Italy, the ancient soul of this tiny nation in the middle of a sea that is embraced by Europe.

“A pilgrimage has meaning if done on foot; it means drawing ever nearer slowly, It is a time-space, not just reaching your goal.

Pilgrimage is about solitude, It is about losing oneself to find oneself again�

â—Ś What is the credential?

Stages of the Way

Fara in Sabina

Greccio Poggio Bustone

Rieti

S. Francis’s Way and S. James’s Way Twins

SAINT JAMES

Saint James’s Way • History • Symbols of the way • The roads to arrive in Santiago de Compostela

Horner

By Fien, Lenja, Lisa and RenĂŠke

ABOUT US -

Belgium Sint-Niklaas Berkenboom Humaniora

Summery 1) 2) 3) 4)

History about Horner Horner’s method Examples of Horner’s method Exercices

William George Horner ° 9 june 1786 British mathematician dad: Ireland Bristol → Kingswood school The Classical seminary 6 daughters, 2 sons

What he wrote... ● ● ● ●

Functional equations Number theory Approximation theory Optics

Most known: Horner’s method

Zoetrope “Daedalum” Pre-film animation devices illusion drawings or photographs

Polynomial

Euclidean tail division

theory

3

-2

1

-1

Diviser x-a a=2

→ Polynomial without x’s

3x2+4x+9 and residue 17 Factorize→ (x-2)(3x2+4x+9) +17

residue

First exercises

calculate these divisions with horner

4X4 -3X3-5X2-7X+3

Number 1

x-2

4 -3 -5 -7 3 together !

2

8 10 10 6 4

5 5

4X3+5X2+5X+3

3

9 residue

Factorize

1 2 -11 2

6

2

8

-6

4

-3

0

X3+2X2-11X+6 X-2

1

= (X-2)(X2+4X-3)

y= -3X3 + 6X2 +15 - 4

X+2 —> a= -2

-3

6

6

-2

3

12

15

-24

-9

-4

18

14

Belgian History: The Gauls Presented by Fre, Lucas en Ken

Game 1: What do you know?

What’s the name of this monument?

What’s our most popular sport?

Can you name these Belgian comics?

What food is typical for Belgium?

Can you name a Belgian football player?

Gaul - Divided in 3 - France, Belgium, Switzerland, The Netherlands, Germany, Italy - Different tribes - Fought a lot against each other - 486 - French Kingdom

Belgica -

Northern part of Gaul Belgica Prima Merovingian Empire Gallia Belgica divided 297 Belgium, The Netherlands

Game 2: Arm Wrestling

Clothing - 12 years old - Admiration - Few clothes

Residence -

Round wall Very strong The roof Partly underground

Weapons - A lot of throwing weapons Example: spear - Shield - Chain mail - Horses, dogs,...

Ireland -

Invasion Brittannia Ireland wasn’t captured Economic tactic

Game 3: Waterpong

Who won?

Congratulations to the winning group!

Thanks for your attention!

t

Diophantus & Algebra

Biography We only know a few things about him:

● He was a greek mathematician. ● He wrote a series of books: Arithmetica. ● He is considered the father of algebra.

Age of Diophantus

His boyhood was one-sixth of his life, One twelfth of it was his youth; then one-seventh of his life passed until he got married.

Five years later he had a son, who died when he was half the age of his father.

Four years after the child´s death, Diophantus passed away.

Age of Diophantus x+x+x+5+x+4=x 6 12 7 2 Least common multiple (6,12,7,2)= = 2x2x3x7= 12x7= 84

6= 2x3 12=2x2x3 7=7 2=2

Age of Diophantus x+x+x+5+x+4=x 6 12 7 2 x+x+x+9+x=x 6 12 7 2 14x + 7x + 12x + 756 + 42x = 84x 14x + 7x + 12x + 42x - 84x = - 756 -9x = -756

Arrange

Least common multiple = 84

Age of Diophantus x+x+x+5+x+4=x 6 12 7 2 x+x+x+9+x=x 6 12 7 2 14x + 7x + 12x + 756 + 42x = 84x 14x + 7x + 12x + 42x - 84x = - 756

-9x = -756

9x = 756 x= 756 9

Age of Diophantus x+x+x+5+x+4=x 6 12 7 2 x+x+x+9+x=x 6 12 7 2 14x + 7x + 12x + 756 + 42x = 84x 14x + 7x + 12x + 42x - 84x = - 756 -9x = -756

9x = 756 x= 756 9

x= 84

Quadratic Equation

12x+x²=3x-18

x²+12x=3x-18

(reorder)

(arrange)

x²+12x-3x+18=3x-18-3x+18

x²+12x-3x+18=0

(equal)

x²+9x+18=0

(solve)

Quadratic Equation 12x+x²=3x-18

First we solve to find the 0

Quadratic Equation 12x+x²=3x-18

First we solve to find the 0

12x+x²-3x+18=0 We do accounts x²+9x+18= 0

Quadratic Equation 12x+x²=3x-18

First we solve to find the 0

12x+x²-3x+18=0 We do accounts x²+9x+18= 0 When we find the 0, we use the formula a=1 b=9 c=18 Substituting...and

Magic Maths 1.

2.

3.

4.

5.

6.

7.

THINK OF A NUMBER. (It should be small as you need to operate with it) THIS NUMBER PLUS 3 MULTIPLY IT BY 2 TAKE 8 AWAY DIVIDE IT BY 2 ADD IT 1 YES! THIS IS YOUR NUMBER

IF YOU HAVE NOT GOT IT, COME ON, YOU SHOULD IMPROVE YOUR MATHS KNOWLEDGE

THERE IS NO MAGIC, JUST MATHS x = YOUR NUMBER A) x B) x+3 C) (x+3)2 D) (x+3)2-8

E) [(x+3)2-8]:2 F) [(x+3)2-8]:2+1

[(x+3)2-8]:2+1 (2x+6-8):2+1 (2x-2):2+1 x-1+1

x

Mathematics Riddle Mathematics Riddle

Mathematics Riddle Mathematics Riddle

Mathematics Riddle Mathematics Riddle 3

2+3=5

2

Mathematics Riddle Mathematics Riddle

Mathematics Riddle Mathematics Riddle

Mathematics Riddle Mathematics Riddle

5

+

3

x

2

Mathematics Riddle Mathematics Riddle

5

+

5

+

3 6

x =

11

2

Thanks for your attention, We hope you learnt some maths ;)

Prehistory

We are natives from Spain, so we will talk about prehistory in our country and we will give some examples of Andalusia, which is in the south of the peninsula.

Anyone knows something about Andalusia?

Andalucia is a region of Spain composed by: Sevilla, Cordoba, Jaen, Malaga, Granada, Cadiz, Almeria and Huelva.

It’s Geographical position was of interest for many species since the beginning of the prehistory.

LET’S MAKE A SMALL QUIZ

What colors does the andalusian flag have?

What dance is typical andalusian and its known throughout Spain?

Flamenco dance was created in andalusia and is known in all Spain.

INDEX

PALEOLITHIC AGE

1.1YEARS 1.2 TOOLS 1.3 EXAMPLES IN ANDALUSIA

NEOLITHIC AGE

2.1 AGES 2.2TOOLS 2.3 EXAMPLES IN ANDALUSIA

AGE OF METALS

3.1 AGES 3.2. TOOLS 3.3. SOCIETY 3.4 EXAMPLES IN ANDALUSIA

PALEOLITHIC AGE

THE ANCIENT HUMAN The Paleolithic has several stages along the human evolved.

4. million years B.C.

2. million

Australopithecus Homo Habilis 3. million

10. thousand B.C.

Homo Erectus Homo Neanderthalensis 230. thousand

Homo Sapiens

THE LIFE IN THE PALEOLITHIC

Their economy was based on hunting , fishing and collection of plant food.

THEIR WEAPONS AND TOOLS The material more used to make tools was the stone , especially the silex.

Examples of the Paleolithic age in Andalusia There are a lot of paleolithic testimonies in Andalusia. Perhaps the most significant is the archaeological richness of the rock of Gibraltar, where we can found the caves where the Neanderthal man´s bones were located for first time.

The rock of Gibraltar

Of this same race there are also remains in the cave of the Gap of Zafarraya (Malaga) and in the Cave of the Carig端ela of Pi単ar (Granada).

Cave of the Gap of Zafarraya

Cave of the Carig端ela of Pi単ar

More examples Other archaeological models of the Paleolithic ones in Andalusia are: - The cave of Ardales: with human feminine and animal representations

- The cave of the Nerja (Malaga): it possesses paintings seals

NEOLITHIC AGE

The neolithic age is consider the final division of the stone age .

Began about 12,000 years ago when the first development of farming appeared in the Epipalaeolithic Near East, and later in other parts of the world.

HOW DID THE LIVE ?

Humans became producers and sedentary

PERMANENT SHELTER ;

PROTECTION FROM WILD ANIMALS

PROTECTION FROM WEATHER

HABILITY TO FORM COMUNITY

POSSIBILITY OF GROW VEGETABLES AND KEEP CARE OF ANIMALS

The results of became sedentary came up with de invention of new professions related to the building of houses , farming ,making tools and clothes .

CAVES

TWO TIPES OF SETTLES HUTS

IN ANDALUSIA

THE TESORO CAVE

NERJA´S CAVE

IN ANDALUSIA

THE CARIHUELA CAVE

METAL AGE

The period of the 3rd, the 2nd, and the 1st millennia BCE was a time of drastic change in Europe.

This has traditionally been defined as the Metal Ages, which may be further divided into stages: the Bronze Age and the Iron Age, which followed a less distinctly defined Copper Age.

INNOVATIONS COOPER

BRONZE

IIRON

The search for raw materials to make new tools gave rise to the beginning of trade activities.

SOCIETY A social division was produced because of the increase of agrarian production by the use of new instruments.

We can study this social division through individual graves in which we found their properties.

METAL AGE IN ANDALUSIA LOS MILLARES (ALMERÍA)

THE END

Equations First and second degree

What is an equation?

An equation is a mathematical equality characterized by having an unknown element, called an incognita.

For example, here we have an equation where the unknown is the x:

2x + 1 = 5 Each side of the equal sign is called a member.

The left side is called the first member and the right side the second member.

The degree of an equation coincides with the highest exponent to which the unknowns are elevated.

First-degree equations First-degree equations are those equations where x is only elevated to 1.

Fill in the box: ‌.. + 3 = 5 Solving equations works in the same way.

In general, any first-degree equation has this form, once simplified:

How to solve first-degree equations?

Practical rules for solving first-degree equations Rule 1: brackets, must be removed as in the usual operations with numbers.

Rule 2: any number can be added or subtracted to or from both members of the equation so any x-term or number that is adding (positive sign) moves to the other side subtracting (negative sign) and vice-versa.

Rule 3: move x-terms to one side and numbers to the other.

Rule 4: combine like terms.

Rule 5: any number that is multiplying, the whole expression, on one side moves to the other side dividing, and vice-versa.

Example of simple first-degree equation First degree equation x is elevated to 1

1) Relocate terms 2) Simplify in groups with similar terms 3) Clear x

How to solve first-degree equations with parentheses Just add one more step to the procedure of solving first-degree equations: Remove parenthesis

How to eliminate parentheses in the first degree equations In front of the parenthesis there may be: a number, a minus sign, or a plus sign.

Multiply the number by all the terms in the parenthesis, taking into account the rule of signs.

Solved exercises with parentheses

How to solve first-degree equations with denominators How to eliminate denominators in a first-degree equation - obtain the common denominator of all the denominators (in this case 24)

- multiply the numerator by its corresponding number, to obtain its equivalent fractions

- in each member, we place everything in a single fraction

- eliminate the denominator directly, multiplying both members by the common denominator We continue to remove parentheses and end up solving the first-degree equation.

3 tricks to easily solve first-degree equations 1. Repeated terms can be crossed out When a term is repeated exactly the same in the two terms, we can cross out those terms.

2. Lesser signs affecting the whole limb can be crossed out If we have a minus sign that affects the whole member, it can be crossed out.

3. Denominators affecting the whole member can be crossed out If the same denominator divides all the first member and all the second member we can cross out them

Second-degree equations Second-degree equations are those where the x appears elevated to 2 in one of its terms.

They can be: -complete second-degree equations; -incomplete second-degree equations.

The values of x that satisfy the equation are called solutions.

Complete second-degree equations They are represented as:

Where a.b and c are the constants of the equations: -a is the number that always goes in front of x square -b is the number that always goes in front of the x -c is the number without unknow A,b and c represent known numbers, with a≠0. If one of these terms missing, we would be talking about incomplete equations.

How to solve complete second-degree equations 1.Identify the constant correctly

2. Applied the following formula to solve the equations:

Let’s see how it is used...

Let’s solve an example:

We have the first equation of second degree in which we identify the constants.

We have to replace the value of each record in the general formula.

We operate within the root taking into account the hierarchy of operations.

We resolve the plus and minus signs and simplify the fractions.

What are Incomplete Second Degree Equations GENERAL FORM OF A COMPLETE SECOND DEGREE EQUATION

The constants b or c, or even both, are missing. We got three types: ● When b=0

● When c=0 ● When b = 0 and c = 0

How to Solve Incomplete Second Degree Equations b=0

EXAMPLE: 1. Clear x² as if it were a first-degree equation:

2. Move the square to the other side as a root, and get a positive and a negative solution:

3. The solutions are 2 and -2.

c=0

EXAMPLE: 1. Draw a common factor, since an x is repeated in both terms.

When a multiplication of two factors results in 0, it means that one of the 2 factors is 0, because any value multiplied by 0 is 0.

2. The solutions are x=0 and x=3.

b=0 and c=0

EXAMPLE:

We only must clear x²: