Square roots-General mistakes

        1-3=4-6 is it correct? yes it is.

             1^2_2x[3/2]x1=2²-2x[3/2]x2    add [3/2]² both side on both side

                   1^2_2x[3/2]x1+[3/2]²=2²-2x[3/2]x2+[3/2]²   identified as a²+2ab+b²=[a+b]²

                         [1-3/2]²=[2-3/2]² cancel the squares on both sides

                              1-3/2=2-3/2

^{:        .}1=2     is it correct ? Actually where is the mistake?

simply we  cancelled the squares on both sides

But it is not correct.

We don’t forget about + on right hand side or left hand side of the equation.

Don't commit mistakes by forgetting fundamentals.

As above, many examples can be taken as a^2_2a[a+b]/2=b^2_2b[a+b]/2 both sides we can get -ab on both sides so satisfied the equation.

Take another example
take a=3 and b=4
then a^2_2a[a+b]/2=b^2_2b[a+b]/2
       3^2_2x3[[3+4]/2]=4^2_2x4[3+4]/2
       9-21=16-28
        -12=-12
Equation satisfied

3^2_2x3[3+4]/2=4^2_2x4[3+4]/2 add {[3+4]/2}² on both sides.
3^2_2x3[3+4]/2+{[3+4]/2}² =4^2_2x4[3+4]/2+{[3+4]/2}² it is in the form of  a²+2ab+b²=[a+b]²
 [3-7/2]²=[4-7/2]²
if cancel the squares on both sides, equation becomes 3=4,which is practically wrong.
3-7/2=-1/2, 4-7/2=1/2, -1/2 is not equal to 1/2 but [-1/2]²=[1/2]²

Square roots Simple method
Any real +ve number in the form of [a+b]²=a²+2ab+b²
+ve numbers are mainly perfect squares and imperfect squares.
step1 identify a number's possibility of perfect square.
All perfect squares have either of the perfect square digit in 1s place as below
i.e 0,1,4,9,6.
256 mayor may not  be perfect square
252 is not perfect square.
456789 mayor may not  be perfect square
789235 is not a perfect square
by this we can estimate a number either chances of perfect square and not.

Finding of Square root of a number

I.e Find the square root of 7225