Here,
root(4)(194-x)+root(4)x=44√194−x+4√x=4
Let, color(red)a=root(4)(194-x)a=4√194−x andcolor(red)
b=root(4)x=>color(red)(a^4)=194-x andcolor(red)( b^4)=xandb=4√x⇒a4=194−xandb4=x
:.color(blue)(a+b=4...to(I) and a^4+b^4=194...to(II)
Squaring (I)=>(a+b)^2=4^2=>a^2+2ab+b^2=16
i.e. color(blue)(a^2+b^2=16-2ab...to(III)
From (II)toa^4+b^4=194,
=>color(brown)((a^2+b^2)^2-2a^2b^2=194
=>(16-2ab)^2-2a^2b^2=194...tousing (III)
=>256-64ab+4a^2b^2-2a^2b^2=194
=>2a^2b^2-64ab+256=194
=>a^2b^2-32ab+128=97
=>a^2b^2-32ab+256=97+128=225
=>(ab-16)^2=15^2
=>ab-16=+-15
=>ab=16+-15=>color(blue)(ab=1 or ab=31
If ab=1,then ,a=1/b to ""color(red)"Please see the comment below"
From (I)to1/b+b=4=>1+b^2=4b
=>b^2-4b+1=0=>b^2-4b+4=3=>(b-2)^2=(sqrt3)^2
=>b-2=+-sqrt3=>b=2+-sqrt3
b^2=(2+-sqrt3)^2=4+-4sqrt3+3=7+-4sqrt3
b^4=(7+-4sqrt3)^2=49+-56sqrt3+48
color(red)(x=97+-56sqrt3
Note:If ab=31=>a=31/b
From (I)to31/b+b=4=>b^2-4b+31=0
triangle=16-124<0=>color(blue)(ab!=31
