N+n²-n⁴+n²-y⁴
I need the value of x in factorise form
someone please help

Recall the equation x²+y²=z² from your school days. We associate this with Pythagoras’ theorem for calculating the sides of a right-angled triangle.

Mathematicians are also interested in natural number (positive integer) solutions to this equation. Such solutions are called Pythagorean Triples.

An example of a Pythagorean Triple is x=3, y=4, z=5 since 3²+4²=5². We can write it more conveniently as (3, 4, 5).

From any Pythagorean Triple, we can construct infinitely many Pythagorean Triples by multiplying through by a number e.g. multiplying (3, 4, 5) by 2 we get (6, 8, 10) which is also a Pythagorean Triple. An exercise for the reader is to show why this always works.

A primitive Pythagorean Triple (x, y, z) is one where x, y, and z are coprime, that is, they have no common factor greater than 1. (3, 4, 5) is primitive.

One challenge is to create a formula for generating Pythagorean Triples.

Here is Euclid’s Formula, where m and n can be any integers of our choice (providing m>n>0) and x, y and z will yield a Pythagorean Triple.

x=2mn

y=m²-n²

z=m²+n²

Example: If m=5 and n=1 then we get the Pythagorean Triple (10, 24, 26). Observe 10²+24²=100+576=676=26².

The following quick check shows that Euclid’s Formula will always work:

x²+y²=4m²n²+(m²-n²)²=4m²n²+m⁴+n⁴-2m²n²=m⁴+n⁴+2m²n²=(m²+n²)²=z²

However, it’s unsatisfying to be given this without understanding where it comes from. Below we will go through a method for deriving Euclid’s Formula from scratch:

Let’s rewrite x²+y²=z² as x²=z²-y²=(z-y)(z+y). This implies x/(z-y)=(z+y)/x.

Note: Division by zero is not an issue because if z=y then this would imply x=0 which is not allowed for a Pythagorean Triple.

However, given x, y and z are positive integers, x/(z-y) is some rational number. Recall the definition of a rational number is any number that can be expressed as m/n where m and n≠0 are integers e.g. 0.5 is a rational number since it’s equal to 1/2.

We can then set x/(z-y)=(z+y)/x=m/n for some integers m and n≠0. We then have the following two equations:

(z-y)/x=n/m — here we took the reciprocal

(z+y)/x=m/n

If we add equations 1 and 2 we get 2z/x=n/m+m/n=(n²+m²)/mn which means we can set x=2mn and z=n²+m².

Similarly, if we subtract equation 2 from 1 then we get 2y/x=m/n-n/m=(m²-n²)/mn which means we can set y=m²-n².

We have then arrived at Euclid’s formula for generating Pythagorean Triples (where m>n>0):

x=2mn

y=m²-n²

z=m²+n²

formula for generating Pythagorean Triples (where m>n>0):

x=2mn

y=m²-n²

z=m²+n²

in mathematics, a function is a relationship between two variables such that the value of one of the variables, called the dependent variable, is determined by the value of the other variable, called the independent variable. there is a specific term we use to describe the set of independent variables of a function.

$450

step-by-step explanation:

first, you have to multiply 150x2=300

next, you have to add 150+300= 450