Converting square root to perfect square

In summary, the conversation was about a problem in which the method of making √(36+x2) a perfect square was used by substituting x with 3( t - 1/t ). The person was asking if there was a rule for choosing this substitution and the expert concluded that there is no specific rule for it.
  • #1
cstvlr
12
0

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.
 
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  • #2
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.
 
  • #3
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 )
 
  • #4
cstvlr said:
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.
 
  • #5
Fair enough, thanks anyway.
 

What is a perfect square?

A perfect square is a number that can be expressed as the product of two equal integers. For example, 9 is a perfect square because it can be written as 3 x 3.

Why do we need to convert square roots to perfect squares?

Converting square roots to perfect squares can help simplify mathematical expressions and make them easier to work with. It can also help in finding the exact value of a square root.

How do you convert a square root to a perfect square?

To convert a square root to a perfect square, you need to find the largest perfect square that is a factor of the number under the square root sign. Then, you can write the square root as the product of the perfect square and another number under the square root sign.

What is the difference between a square root and a perfect square?

A square root is a number that, when multiplied by itself, gives the original number. A perfect square is a number that can be expressed as the product of two equal integers. So, while all perfect squares have a square root, not all square roots are perfect squares.

Can you convert any square root to a perfect square?

No, not all square roots can be converted to perfect squares. Only square roots of numbers that are perfect squares themselves can be converted to perfect squares. For example, the square root of 16 can be written as 4 x √1, but the square root of 10 cannot be simplified further.

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