How to find quickly square root of a perfect square polynomial of degree 6?

How to find quickly square root of a perfect square polynomial of degree 6? This is a little method which will enable you to get the answer quickly. Let the polynomail be Px^6+Qx^5+Rx^4+Sx^3+Tx^2+Ux+V. Let its root be ax^3+bx^2+cx+d. a= square root of P. b= Q/2a c= (R-b^2)/2a d= U/2c

Coordinate geometry trick: If co-ordinates of two points on a line are given and you are asked to find the equation of the line. How to find it quickly.

 Coordinate geometry trick:

If co-ordinates of two points on a line are given and you are asked to find the equation of the line. How to find it quickly.

0) If Y2/X2 = Y1/X1 = k

the required equation is y = kx.

or kx-y=0.

Example 1:

A (2,4)

B (1,2)

Here it is quite easy to understand that k = 4/2 = 2/1 = 2.

so y = 2x

or 2x-y=0 is the required equation of the line.

*Or*

1) Write first the coordinates of that point whose y- co-ordinate is larger than y-co-ordinate of the other point.

2) Below it, write the co-ordinates of the other point.

3) Find the difference between the Y- co-ordinates. You have got now the coefficient of the x in the required equation of the line.

4) find the negative of the difference between the X coordinates in the same order. This is the coefficient of y in the required equation of the line.

5) Now X2 , Y2

              X1 , Y1.

Find + ( X2 Y1 - X1 Y2 ).

It is the constant part of the equation.

5a) write the answers obtained steps 3), 4) and 5) with x, y and appropriate signs for the constant, and equate it to zero.

6) This is the required equation of the line in a x + b y + c = 0 form. *Refer Example 2 given below.*

7) If in a x + b y + c = 0, a, b & c not coprimes, then divide the equation by the highest common factor (HCF) of a,b,c.

This is your required equation. *Refer Example 3 given below.*

Example 2:

A(3,-2)

B(4,-4)

-2>-4 

A(3,-2)

B(4,-4)

[-2-(-4)] x - (3-4) y + (3x-4 - 4x-2) =0

2x+y+(-12+8)=0

2x+y=4.

is the required equation.

Example 3:

B (1,-1)

A (7,7)

7 > -1

A (7,7)

B (1,-1)

[(7 - (-1) ] x -(7 - 1)y + (7 x - 1 - 7 x 1) = 0

8X - 6y -14 =0

8, 6 and 14 are not coprimes and their HCF is 2 therefore dividing the equation by 2 we get,

4x -3y - 7 = 0

or 4x -3y = 7 is the required equation of the line passing through the given points.

Hope you liked the post. It is more useful for 11 Std Science student who are preparing for CET or JEE.

How to form sums based on Apollonius theorem in which 3 sides of a triangle and a median all are natural numbers?

1) We know that, an odd number's square will be odd. Adding two odd numbers square will give you even number. So adding squares of two odd numbers will give you an even number.

2) Take 19 and other prime number.

3) Add their squares. Divide it by 2.

4) Find in how many ways the last digit can be formed. 

5) Try to express the number obtained in step 3 as square of two numbers. You may write natural numbers and their squares separately.

6) Remember 2,3,7 and 8 do not come at the end of (unit place of) a perfect square.

7) Discard those possibilities.

8) 1,3,5,7 and 9 can come at / as last digit in following ways.

1: 0 & 1 or 5 & 6

3: 4 & 9

5: 1 & 4; 5 & 0 or 6 & 9

7: 1 & 6

9: 4 & 5 or 0 & 9.

For above mentioned step 5) consider natural numbers up to square root of the largest perfect square less than or equal the sum obtained step 3.

9) We know that sum of two sides of a triangle is greater than third side.

Check whether all the three triangles so formed satisfy this condition.

10) You can create another sum. If three sides of larger triangle are obtained as a-b-2d and median on side 2d as c; the other sum (triangle) is a-b-2c and its median will be d.

- Thursday, the midnight of 22 Nov 18 2:12 a.m. 23 Nov 2018.

3 Prostrations to Shree Saibaba, Shirdi, Dist. Ahmednagar, Maharashtra, India!!!

For example:

In triangle ABC, sides are as given below. Find length of median AM on side BC.

1) AB=13, AC=29, BC=24.

2) AB=13, AC=29, BC=38.

Graphical presentation of "a square minus b square".

Mathematrix Tutorials

My Tuition Classes

How to find whether / if a given number is prime or not?

How to find whether a given number is prime or not?
1) Find largest perfect square less than or equal to this number. Find square root of this perfect square.
2) If the given number is very big you can use long division method of finding square root.
3) Apply divisibility tests for all prime numbers less than or equal to this number. If you do not know divisibility tests for numbers, divide the given number by those prime numbers.
if all the tests fail, the given number is prime number.