rationals

[ionic pulsar]

Couldn’t we solve it by the second method aswell?Where is the mistake

[hollow fossil]

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[frail owl]

via the second method you're squaring each part separately, which you can't do:
to complete your second method correctly you would have to do:
(√5+7)²=5+49+14√5=54+14√5≤(6+4√5)²=36+80+48√5=116+48√5...

which takes a whole lot longer

[ionic pulsar]

So squaring the whole side.Unlike multiplication it can’t be done seperately

[frail owl]

what? whatever you do to one side you must do to the other side

[ionic pulsar]

So it would lead to this

Leading to 13=24 root s

It’s an s not 5