If 2x+ 2/sqrt x =7 Then x - 2sqrt x = ?

If #2x+ 2/sqrtx =7 " Then "x - 2sqrt x = #?

1 Answer
Guillaume L.
May 26, 2018

#x_1=(-1+sqrt2)/2#
#x_2=-(1+sqrt2)/2#

Explanation:

#2x+2/sqrtx=7#

#2x+(2sqrtx)/x-7=0#

#(2x²-7x+2sqrtx)/x=0#

#2x²-7x+2sqrtx=0#

Let #X=sqrtx#

#2X⁴-7X²+2X=0#

#2X³-7X+2=0#

#2X³-8X+X+2=0#

#color(red)(2X³+4X²)color(blue)(-4X²-8X)+X+2=0#

#color(red)(2X²(X+2))color(blue)(-4X(X+2))+(X+2)=0#

#(X+2)(2X²-4X+1)=0#
So:

#X=-2 => cancel(sqrtx=2)# impossible.
Or:

#2X²-4X+1=0#

#Delta=(-4)²-4*2*1#

#Delta=8#
So
#X_1=(4-sqrt8)/4=(2-sqrt2)/2#

#X_2=(4+sqrt8)/4=(2+sqrt2)/2#

So : #x_1=(3-sqrt2)/2#
#x_2=(3+sqrt2)/2#
So :
#x-2sqrtx=(3+sqrt2)/2-2*(2+sqrt2)/2#
#=-(1+sqrt2)/2#
Or #=(3-sqrt2)/2-2*(2-sqrt2)/2#
#=(-1+sqrt2)/2#
\0/ here's our answer!

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