Converting square root to perfect square

[cstvlr]

Homework Statement

Hi,

I have a problem in my book in which they use a method of making sqrt( 36 + x^2 ) a perfect square by simply making x = 3( t - 1/t ) and then we get 9( t + 1/t )^2 by substituting back into sqrt(36 + x^2). My question is that why did the chose 3( t - 1/t ), is there a rule?

thanks in advance.

 

[rock.freak667]

What were they trying to do with the expression √(36+x²) ? For example, if they were trying to integrate, you could have made a simple trig substitution instead of trying to think of some odd set of functions to string together.

 

[cstvlr]

no its in an equation form,

C(x) = [(9-x) + 1.25sqrt( x^2 + 36 )], where C(x) is a function.

and they're trying to solve for x, so they used the method above, and I know how to solve it, but I lost it when they used let x = 3( t - 1/t )

 

[rock.freak667]

no its in an equation form,

C(x) = [(9-x) + 1.25sqrt( x^2 + 36 )], where C(x) is a function.

and they're trying to solve for x, so they used the method above, and I know how to solve it, but I lost it when they used let x = 3( t - 1/t )

x = 3( t - 1/t ) ⇒ x² = 9( t - 1/t )²

∴√(36+x²)=√(36+9(t- 1/t)²), how does that turn into an expression like √(a+b)²?

I do not think there rule per se for using that substitution though.

 

[cstvlr]

Fair enough, thanks anyway.