Question

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`\sqrt{19 + \sqrt{30 + √(32 + x ) } } = 5`
can someone please solve this​

Answer 1

We'll have to repeatedly square both sides of the equation, in order to get rid of the square roots. Squaring a first time yields

`19+\sqrt{30+√(32+x)}=25`

Move the 19 to the right hand side:

`\sqrt{30+√(32+x)}=6`

And square again:

`30+√(32+x)=36 \iff √(32+x)=6`

Square one last time:

`32+x=36 \iff x=36-32=4`

Let's check the solutions: all these squaring might have created external solutions:

`\sqrt{19+\sqrt{30+√(32+4)}}=\sqrt{19+√(30+6)}=√(19+6)=√(25)=5`

So,
`x=4` is a feasible solution.