Is √x² = x or -x or none.
Now to solve this situation, always remember in mind. √X is always a positive number where X is a real number
say for e.g. √25 = 5 a +ve number. √(-5)2 = 5 or √(5)2 = 5.
√36 = 6 a +ve number.
similarly √x² is always a +ve number.
So why can’t we write √x² = x ????? –> NO
Think before you write. Do you know that x is positive or negative. NO. but you know that √x² is a positive number.
positive number =/= negative number.
For e.g. x=-5 , so √(-5)² =/= -5
but √(-5)² = 5 = -(-5) =-x
So can we write √x² = -x ????? —> NO
Again think. what will happen if x is a positive number.
For e.g. x=5 , so √(5)² =/= -5
but if x=5, √(5)² = 5 =x.
So what is this √x² basically.
Going through the above examples again we find that, the magnitude of the final result is same but there is a difference in sign (+ or -) depending on the value of x.
so to make √x² compatible with RHS, we will have to make RHS always +ve.
So √x² = |x| , which is always true.
where |x| is absolute value or modulus of x
and √x² = |x| = x if x >0 and -x if x<0.
If you feel more points could/should be added. Please add in comments , we would edit the post as required.