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[BASIC Algebra] Distributing a root over underlying terms?

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Let x = (A^2 +B^2 -C^2 )/(2AB)

The expression is: √(1 - x^2 )

The other expression is: √(1) - √(x^2 ) = 1 - √(x^2 )

If the expressions are equal, then:

√(1-x^2 ) = 1 - √(x^2 )

1 - x^2 = 1 - 2√(x^2 ) + x^2

1 = 1 - 2√(x^2 ) + 2x^2

2√(x^2 ) = 2x^2

√(x^2 ) = x^2

If x ≥ 0:

x = x^2

x^2 - x = 0

x(x-1) = 0

x = 0 or x = 1

If x < 0:

-x = x^2

x^2 + x = 0

x(x+1) = 0

x = -1

Therefore: x = 0, x = -1, or x = 1.

Therefore, the equation is not always true.

The expression can be factorised: √(1-x^2 ) = √((1-x)(1+x)) = √(1-x) √(1+x)
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sqrt(a+b) is not equal to sqrt(a)+sqrt(b)

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