Question

How do you solve `x+1/x=7`?

Answer 1

`color(red)(x=3.85 or 0.15`

Explanation:

`x+1/x=7`

or, `x+1/x-7=0`

or, `(x^2+1-7x)/x=0`

or, `x^2-7x+1=0`

or, `x=(-(-7)+-sqrt((7^2)-4*1))/(2*1)=(7+-sqrt45)/2=(7+-3sqrt5)/2`

`therefore x=3.5+1.5sqrt5 or 3.5-1.5sqrt5`

or, `x=3.5+3.35 or 3.5-3.35`

or, `x=3.85 or 0.15`

Answer 2

`x=7/2+-3/2sqrt5`

Explanation:

`"multiply through by "x`

`x^2+1=7x`

`"arrange in "color(blue)"standard form ";ax^2+bx+c=0`

`"subtract "7x" from both sides"`

`x^2-7x+1=0`

`"check the value of the "color(blue)"discriminant"`

`•color(white)(x)Delta=b^2-4ac`

`"with "a=1,b=-7" and "c=1`

`b^2-4ac=(-7)^2-(4xx1xx1)=45`

`"thus the roots are real and irrational"`

`"solve using the "color(blue)"quadratic formula"`

`•color(white)(x)x=(-b+-sqrt(b^2-4ac))/(2a)`

`x=(7+-sqrt45)/2=(7+-3sqrt5)/2`

`x=7/2+-3/2sqrt5larrcolor(red)"exact solutions"`