A positive real number is one more than its reciprocal. Find the number.

Question

Is the equation x=1/x + 1

Answer 1

Mary Anne,

 

You are correct that the initial equation is x = 1/x + 1

 

You then rearrange the equation into a quadratic by

 

1. Multiply both sides of the equation by x to get x^2 = 1 + x

2. Subtract x and 1 from both sides to get x^2 - x - 1 = 0

3. Using the quadratic formula (where a=1, b=-1, c=-1), you find that possible values for x are (1 + sqrt(5)) / 2 and (1 - sqrt(5)) / 2

4. Since the answer must be positive and real, then x = (1 + sqrt(5)) / 2 (or 1.618) is your solution.

Answer 2

x = 1 + 1/x

 

Multiply both sides by x:

 

x² = x + 1

x² - x - 1 = 0

 

Use the quadratic formula:

 

x = 1/2 ± (1/2)√((-1)2-4(1)(-1))

x = (1±√5)/2

 

Now √5 >1, so x is only positive when it's :

 

x = (1+√5)/2

 

Now let's check if it's right:

 

x = 1/x + 1

x - 1 = 1/x

 

Substitute (1+√5)/2 for x:

 

(1+√5)/2 - (2/2) = 2/(1+√5)

 

Multiply the numerator and denominator of the RHS by the conjugate of the denominator:

 

(√5 - 1)/2 = (2/(1+√5))*(1-√5)/(1-√5)     

 

(√5 - 1)/2 = 2*(1-√5)/-4

 

(√5 - 1)/2 = (√5 - 1)/2

 

Yup, it seems to be true!

 

 

Answer 3

x=1/x+1 becomes x-1=1/x which is then x(x-1)=1

 

factoring we get x^2-x=1

 

completing the square we get (x-.5)^2 -.25=1

 

therefore, (x-.5)^2=1.25

 

and now we take the positive sq root of both sides to get (x-.5)= sqrt(1.25)

 

thus x=.5+sqrt(1.25)

 

or x=1.61803