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Find the value of x satisfying the equation \[ \sqrt{ 3\sqrt{x+5} - \sqrt{x-2} } = 3. \]

 Jan 30, 2021
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Let's neglect negative square roots. 

 

Square both sides so that what we have under the big radical equals 9. 

 

So we've got          3•sqrt(x+5) – sqrt(x–2)  =  9  

 

The answer comes out even so each of the sqrts has to come out even.  

 

By a lot of contemplation and a little brute force, I found that x = 11 will work splendidly.  

 

All I did was find a number that would form squares when you add 5 to it and when you subtract 2 from it. 

 

It really wasn't all that hard to do. 

 

Check:  3•sqrt(11+5) – sqrt(11–2)   =   3•sqrt(16) – sqrt(9)   =   3•4 – 3   =   12 – 3  = 9 

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 Jan 30, 2021

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