Question

If `(x+6)/x^(1/2) = 35` then what is the value of `(x+1)/x` ?

Answer 1

1

Explanation:

Solve for x:

`(x+6)/x^(1/2)=35`
`x+6=35x^(1/2)`

I chose to square both sides in order to get rid of the square root.

`(x+6)^2=1225x`
`x^2+12x+36=1225x`
`x^2-1213x+36=0`

I don't think I can factor this, so I'm going to apply the quadratic formula instead!

`x=(-b+-sqrt(b^2-4ac))/(2a)`
`x=(1213+-5sqrt(58849))/2`
`x=(1213+5sqrt(58849))/2` because `(((1213+5sqrt(58849))/2)+6)/sqrt((1213+5sqrt(58849))/2)=35`

Now all you have to do is plug `x=(1213+5sqrt(58849))/2` into `(x+1)/x`!

`(x+1)/x~~1`

Answer 2

`(x+1)/x = 1285/72+-35/72sqrt(1201)`

Explanation:

Given:

`(x+6)/x^(1/2) = 35`

Multiply both sides by `x^(1/2)` to get:

`x+6 = 35x^(1/2)`

Square both sides to get:

`x^2+12x+36 = 1225x`

Subtract `1225x` from both sides to get:

`x^2-1213x+36 = 0`

Next note that we want to find:

`(x+1)/x = 1+1/x`

Multiplying the quadratic we have found by `1/x^2` we get:

`36(1/x)^2-1213(1/x)+1 = 0`

So by the quadratic formula we find:

`1/x = (1213+-sqrt((-1213)^2-4(36)(1)))/(2(36))`

`color(white)(1/x) = (1213+-sqrt(1471369-144))/72`

`color(white)(1/x) = (1213+-sqrt(1471225))/72`

`color(white)(1/x) = (1213+-35sqrt(1201))/72`

So:

`(x+1)/x = 1+1/x = 1285/72+-35/72sqrt(1201)`