Equations

An equation is a mathematical statement that has two expressions separated by an equal sign. Solving an equation means manipulating the expressions and finding the value of the variables.

An equation in one variable has a single unknown quantity called a variable represented by a letter. Eg: ‘x’, where ‘x’ is always to the power of 1. This means there is no ‘ x² ’ or ‘ x³ ’ in the equation.

5x + 2 = 2x + 17
Subtract 2x from both sides:
5x + 2 - 2x = 2x + 17 - 2x
Simplify both sides:
3x + 2 = 17
Subtract 2 from both sides:
3x + 2 - 2 = 17 - 2
Simplify both sides:
3x = 15
Divide both sides by 3:
Simplify both sides:
x = 5

Zero Product

(3x - 2)(x + 1) = 0

Use the principle of zero product, which says, if ab = 0 , either a, b, or both must be equal to zero.

3x - 2 = 0 0x + 1 = 0
3x = 2 x = -1
x = (2/3)



The solutions are : -1 and 2/3

Quadratic equation

A general quadratic equation can be written in the form, where x represents a variable or an unknown and a, b, and c are constants with a ≠ 0. (If a = 0, the equation is a linear equation .)

Polynomial equation solved by factoring :

3 − 11/2 x − 5x² = 0
5x² +11/2 x − 3= 0
5x² +11/2 x− 3 = 0
(5x − 2 )(2x + 3)= 0
So , the solutions are 2/5and −3/2

Solved by completing the square

Completing the square is a technique for converting a quadratic polynomial of the form

Ax^2+bx+c to a(x-h)^2+k

3x²+12x+27 = 3(x²+4x+9 )
= 3((x+2)²+5)
=3(x+2)²+5

By Parameswari & Sivapriya 2nde 2 =)

The Intercept theorem

Thales was a great Greek philosopher and mathematician. It is said that he found this theorem when he tried to calculate the height of pyramids in Egypt. His theorem states than in a triangle, if a line is parallel to one side of a triangle then it divides the other two sides proportionally. Now let's not waste any time and get ready for the demonstration !

Euclid is renowned for writing The Elements, in thirteen books he tried to compile all the mathematic definitions, theorems and proof. He also wrote the proof of the intercept theorem, in his 6th book. So let’s get started:

We have a triangle ODC with (AB)//(DC), now watch:

• AADC= AAID+ ADIC
• ABDC=ABIC+ADIC
• AAID+ ADIC= ABIC+ADIC
• AAID= ABIC

Now let’s continue, let’s call h, the height issued from B
AOAB= (OA*h)/2
AODB= (OD*h)/2
AOAB/ AODB= OA/OD
In the same way, k is the height issued from A.
AOAB= (OB*k)/2
AOAC= (OC*k)/2
AOAB/ AOAC= OB/OC
AODB=ADIA+AOAIB=ABIC+AOAIB=AOAC
We can conclude that: OA/OD=OB/OC

And that's it !

Are you saying that we forgot something? Hmm, you're
right, it's not over yet, we didn't talk about AB/DC, thanks
for remainding us.

Let’s continue, we have (AE)//(OC).
If we use the same method as earlier, we will get:
DA/DO=DE/DC
AO/DO=EC/DC
And ABCE is a parallelogram so we have: EC/DC=AB/DC
OA/OD=AB/DC=OB/OC
And that’s all!

Thanks for watching!

A project by Guillaume LARROUTUROU & Ismaïl RAZACK.

Square root of two, an irrational number: how to prove it?

√2=1.41421356237309504880168872420969807856967187537694807317667973799...

And it continues!!
What is a rational number?
It is a number which can be expressed in the form a/b, a fraction in its lowest terms, where a and b are two integers (whole numbers), and b different from 0 and 1.
And expressing √2 in this form is not obvious!
So now we're going to prove that it's irrational, by contradiction, that means:
If it was a rational number, then

√2=a/b

2=a²/b²
2b²=a²
Now we know that a² is even, because it is a multiple of 2, and we can deduce that a is also even! Why?
If a is even, then it equals to a multiple of two: a=2 x k
a²=a x a

substituting a by 2k
a²=2k x 2k
which gives us

a²=4k²
a² is also even!

If the product of 2 whole numbers has to be even, then at least one of those numbers has to be even.So if the square of a number (which is a multiplication of the number by itself) is even, that number is also even.We deduce that if a² is even than a is even.

When it comes to an odd number, the product of 2 numbers which are not multiples of 2, the product can't be even too.

So now that we know that a is even if √2 is rational, it equals to a/b and we know that a equals to 2k, so we can say that:

√2 = 2k/b
then

2 = (2k)²/b²

2b² = 4k²

b² = 2k²

Being a multiple of 2, b² is even, in the same way, b is also even.
But we have started the process, saying that a/b is in its lowest terms but they are both even, which means that they can be simplified by two.

So we can deduce that √2 is an irrational number.
By ADY Canimozhi and SIVAPRAGASSAM Aswin