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

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

1 Answer
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
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