Question

How do you solve the equation: `x + 1/x+1 = 4`?

Answer 1

`x=\frac{3\pm sqrt{5}}{2}\approx 2.618, 0.382`

Explanation:

Two methods:

1) Multiply everything by `x` first to get `x^2+1+x=4x`, or `x^2-3x+1=0`. The quadratic formula now gives `x=\frac{3\pm sqrt{9-4}}{2}=\frac{3\pm sqrt{5}}{2}\approx 2.618, 0.382`.

2) Add the fractions on the left first by getting a common denominator of `x`:

`x^2/x+1/x+x/x=4\Rightarrow (x^2+x+1)/x=4/1`

Now "cross-multiply" to get `x^2+x+1=4x`, or `x^2-3x+1=0`. This gives the same answer now as Method 1: `x=\frac{3\pm sqrt{9-4}}{2}=\frac{3\pm sqrt{5}}{2}\approx 2.618, 0.382`.

As with all algebra problems like this, you should check the answers in the original equation. If you try this with, for example, `x=(3+sqrt{5})/2` in exact form, you'll get:

`x+1/x+1=(3+sqrt{5})/2+2/(3+sqrt{5})+1`

`=((3+sqrt{5})(3+sqrt{5})+2*2+2(3+sqrt{5}))/(2(3+sqrt{5}))`

`=(9+6sqrt{5}+5+4+6+2sqrt{5})/(6+2sqrt{5})=(24+8sqrt{5})/(6+2sqrt{5})`

`=(4(6+2sqrt{5}))/(6+2sqrt{5})=(4cancel((6+2sqrt{5})))/(cancel(6+2sqrt{5}))=4`

Answer 2

Multiply all terms by `x`; rearrange and solve as a standard quadratic to get:
`color(white)("XXXX")` `x = (3+-sqrt(5))/2`

Explanation:

`x + 1/x + 1 = 4`

`rArr` `color(white)("XXXX")` `x^2+1+1x = 4x`

`rArr` `color(white)("XXXX")` `x^2-3x+1 = 0`

using the quadratic formula:
`rArr` `color(white)("XXXX")` `x = (3+-sqrt((-3)^2-4(1)(1)))/(2(1)`

`color(white)("XXXX")` `color(white)("XXXX")` `x = (3+-sqrt(5))/2`