V. Find the intercepts of y equals x square minus one and x square plus y square equals thirty. We know that y equals x square minus one is a curve which this shape is parabola and x square plus y square equals thirty is a circle equation.


x square plus y square equals thirty is a circle with radius is root of thirty.
x square equals to thirty minus y square.
x equals to the root of thirty minus y square in bracket.

Equation substitute in square equation.

Y equals x square minus one.
Y equals the root of thirty minus y square in bracket square minus one.
Y equals thirty minus y square minus one.
Y equals twenty nine minus y square.
y square plus y minus twenty nine equals zero.

We know that this is square equation, and then we will find the value of y. for find the value of y, we use of a, b, c formula.

Y one and y two equals open bracket negative b plus minus the root of b square minus four times a times c in bracket close bracket over two times a.
Y one and y two equals open bracket negative one plus minus the root of one minus four times one times negative twenty nine in bracket close bracket over two.
Y one and y two equals open bracket negative one plus minus the root of one hundred and seventeen in bracket close bracket over two.


Y one equals open bracket negative one plus the root of one hundred and seventeen in bracket close bracket over two.
Y one equals negative one plus ten point eight in bracket over two.
Y one equals four point nine.


The value of x one is:
X one equals the root of thirty minus four point nine square in bracket.
X one equals the root of thirty minus twenty four point oh one in bracket.
X one equals the root of five point nine.
X one equals two point four.


Y two equals open bracket negative one minus the root of one hundred and seventeen in bracket close bracket over two.
Y two equals open bracket negative one minus the root of ten point eight close bracket over two.
Y two equals negative eleven point eight over two.
Y two equals negative five point nine.



The value of x two is:
X two equals the root of thirty minus negative five point nine square in bracket.
X two equals the root of thirty minus thirty four point eighty one in bracket.
X two equals the root of negative four point eighty one.
This is not value for x two.

So the intercept from two equations is in x equals two point four and y equals four point nine.




IV. Showing that the root of 2 is irrational number

Root of 2 is the number irrasional first introduced by the Greeks. They said that the long diagonal of a triangular carpenter's square with the length of two sides that form a right angle-right angle to each one may not rational (irrasional).
Phytagoras obtained from the theorems that the length of hypotenuse (hipotenus) triangle is the root of 2.
The diagonal distance of square whose sides one is the root of one square plus one square in bracket equals the root of two. So diagonal distance is not representative rational number. We need the root of two. Itsmean the root of two equals a over b, where a over b is the prime number.