Question

How this is possible ???????
If two different variables are equal then their value(two different numbers) are also equal

So how two different numbers can be equal
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Answer 1

Before answering this question, we must know 'what does the symbol √ actually mean.'

The symbol √ is used to represent the "positive square root" of a number, not all the square roots.

To represent all the square roots we must show it as '±√'.

E.g.: √4 = 2 while ±√4 = ±2

And even though,

√4 = √[(-2)²] = 2

Algebraically, it's always,

√(x²) = |x|

Now come to the question.

√(a² + b² - 2ab) = √[(a - b)²] = √[(b - a)²]

So it's actually equal to,

√(a² + b² - 2ab) = |a - b|

Or

√(a² + b² - 2ab) = |b - a|

Thus,

a - b = - (b - a)

What's wrong with that solution is given below.

Equation (1) showing √(a² + b² - 2ab) = √[(a - b)²] = a - b is an assumed or confirmed matter.

This also shows,

√(a² + b² - 2ab) = √[(a - b)²] = |a - b| = a - b

Hence we can say that "a - b is positive here."

This implies "b - a is negative here."

From (1) it implies (2) showing √(a² + b² - 2ab) = √(b² + a² - 2ab) = √[(b - a)²] = |a - b|, thereby actually implying -(b - a) = a - b, not b - a directly, since a - b is considered.

Here it also implies that, a > b.

This is case 1.

Case 2 shows that "b - a is positive."

Thus (2) will be √(a² + b² - 2ab) = √(b² + a² - 2ab) = √[(b - a)²] = b - a.

Since b - a is positive, a - b will be negative.

So, (1) changes to √(a² + b² - 2ab) = √[(a - b)²] = |a - b| = -(a - b).

Case 2 also implies that, b > a.

Case 1 solution is given as the question. Even √(a² + b² - 2ab) from (1) changes to √(b² + a² - 2ab) in (2), the value doesn't change, so a - b will be positive in both equations. It can be found out by giving values for a and b as examples.

And also, there is case 3, which implies a = b and a - b = b - a if and only if a² + b² - 2ab = 0.