Question

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

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

Answer 1

`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!