A math puzzle that had stumped mathematicians for decades was finally solved. It’s called a Diophantine Equation, and it’s sometimes known as the “summing of three cubes”: Find x, y, and z such that x³+y³+z³=k, for each k from 1 to 100.

On the surface, it seems easy. Can you think of the integers for x, y, and z so that x³+y³+z³=8? Sure. One answer is x = 1, y = -1, and z = 2. But what about the integers for x, y, and z so that x³+y³+z³=42?

That turned out to be much harder—as in, no one was able to solve for those integers for 65 years until a supercomputer finally came up with the solution to 42. (For the record: x = -80538738812075974, y = 80435758145817515, and z = 12602123297335631. Obviously.)

That’s the beauty of math

Q.
alpha and beta are the zeros of x^2-4x-4 ,
Then find the value of 1/alpha - 1/beta
Solution :-
     

Q. If P is a prime prove that √P is irrational.
Solution:-
           Let √P be rational
    √P = a/b  (a,b are co-prime),b is not equal to 0
    a = √Pb
    Squaring both sides
   a^2 = Pb^2.............(1)
  P divides a^2
  P divides a
  a = PR ( for some integer R)
  Put in (1)
  P^2R^2 = Pb^2
  PR^2 = b^2
  P  divides b^2
  P divides b
  b = PQ (for some integer Q)
  a = PR , b = PQ
But a,b are co-prime
This contradicts
Therefore,√P is irrational

Quadratic formula

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A quadratic function with roots x = 1 and x = 4.

In algebra, the quadratic formula is the solution of the quadratic equation. There are other ways to solve the quadratic equation instead of using the quadratic formula, such as factoring, completing the square, or graphing. Using the quadratic formula is often the most convenient way.

The general quadratic equation is

${\displaystyle ax^{2}+bx+c=0\ \ .}$

Here x represents an unknown, while a, b, and c are constants with a not equal to 0. One can verify that the quadratic formula satisfies the quadratic equation by inserting the former into the latter. With the above parameterization, the quadratic formula is:

$${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}\ \ }$$

Each of the solutions given by the quadratic formula is called a root of the quadratic equation. Geometrically, these roots represent the x values at which any parabola, explicitly given as y = ax² + bx + c, crosses the x-axis. As well as being a formula that will yield the zeros of any parabola, the quadratic formula will give the axis of symmetry of the parabola, and it can be used to immediately determine how many real zeros the quadratic equation has.

Derivations of the formulaEdit

By using the 'completing the square' methodEdit

The quadratic formula can be derived with a simple application of technique of completing the square
 The  derivations are as follows:

DerivationEdit

Divide the quadratic equation by a, which is allowed because a is non-zero:

${\displaystyle x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}=0\ \ .}$

Subtract c/a from both sides of the equation, yielding:

${\displaystyle x^{2}+{\frac {b}{a}}x=-{\frac {c}{a}}\ \ .}$

The quadratic equation is now in a form to which the method of completing the square can be applied. Thus, add a constant to both sides of the equation such that the left hand side becomes a complete square.

${\displaystyle x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}=-{\frac {c}{a}}+\left({\frac {b}{2a}}\right)^{2}\ \ ,}$

which produces:

${\displaystyle \left(x+{\frac {b}{2a}}\right)^{2}=-{\frac {c}{a}}+{\frac {b^{2}}{4a^{2}}}\ \ .}$

Accordingly, after rearranging the terms on the right hand side to have a common denominator, we obtain:

${\displaystyle \left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}-4ac}{4a^{2}}}\ \ .}$

The square has thus been completed. Taking the square root of both sides yields the following equation:

${\displaystyle x+{\frac {b}{2a}}=\pm {\frac {\sqrt {b^{2}-4ac\ }}{2a}}\ \ .}$

Isolating x gives the quadratic formula:

${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}\ \ .}$

The plus-minus symbol indicates that

${\displaystyle x_{1}={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\quad {\text{and}}\quad x_{2}={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}}$

                Who was Pythagoras

Pythagoras of Samos ( 570 – 495 BC)was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna graecia and influenced the philosophies of plato, Aristotle, and, through them, western philosophy. Knowledge of his life is clouded by legend, but he appears to have been the son of Mnesarchus, a seal engraver on the island of samos. Modern scholars disagree regarding Pythagoras's education and influences, but they do agree that, around 530 BC, he travelled to croton, where he founded a school in which initiates were sworn to secrecy and lived a communal, ascetic lifestyle This lifestyle entailed a number of dietary prohibitions, traditionally said to have included vegetarianism, although modern scholars doubt that he ever advocated for complete vegetarianism.

Pythagoras
Bust of Pythagoras of Samos in the Capitoline Museums, Rome[1]
Born	c. 570 BC Samos
Died	c. 495 BC (aged around 75) either Croton or Metapontum
Era	Ancient Greek philosophy
Region	Western philosophy
School	Pythagoreanism
Main interests	Ethics Mathematics Metaphysics Music Mysticism Politics Religion
Notable ideas	Communalism Metempsychosis Musica universalis Attributed ideas: Five climactic zones Five regular solids Proportions Pythagorean theorem Pythagorean tuning Sphericity of the Earth Vegetarianism
Influences Thales Anaximander Pherecydes Themistoclea
Influenced Plato and, through him, all of Western philosophy

The teaching most securely identified with Pythagoras is metempsychosis, or the "transmigration of souls", which holds that every soul is immortal and, upon death, enters into a new body. He may have also devised the doctrine of musica universalis, which holds that the planets move according to mathematical equations and thus resonate to produce an inaudible symphony of music. Scholars debate whether Pythagoras developed the numerological and musical teachings attributed to him, or if those teachings were developed by his later followers, particularly philolaus of croton . Following Croton's decisive victory over sybaris in around 510 BC, Pythagoras's followers came into conflict with supporters of democracy and Pythagorean meeting houses were burned. Pythagoras may have been killed during this persecution, or escaped to metapontum, where he eventually died.

In antiquity, Pythagoras was credited with many mathematical and scientific discoveries, including the Pythagorean theorem, Pythagorean tuning, the five regular solids , the theory of proportions, the sphericity of the earth, and the identity of the morning and evening stars as the planet Venus. It was said that he was the first man to call himself a philosopher ("lover of wisdom") and that he was the first to divide the globe into five climatic zones . Classical historians debate whether Pythagoras made these discoveries, and many of the accomplishments credited to him likely originated earlier or were made by his colleagues or successors. Some accounts mention that the philosophy associated with Pythagoras was related to mathematics and that numbers were important, but it is debated to what extent, if at all, he actually contributed to mathematics or natural philosophy.

Pythagoras influenced Plato, whose dialogues, especially his timaeus, exhibit Pythagorean teachings. Pythagorean ideas on mathematical perfection also impacted ancient Greek art. His teachings underwent a major revival in the first century BC among middle Platonists, coinciding with the rise of Neopythagoreanism. Pythagoras continued to be regarded as a great philosopher throughout the middle ages and his philosophy had a major impact on scientists such as Nicolaus Copernicus, Johannes Kepler, and Issac Newton. Pythagorean symbolism was used throughout early modern European  esotericism and his teachings as portrayed in ovid's Metamorphoses influenced the modern vegetarian movement.