positive real number

Question

Find a positive real number that is 2 smaller than its reciprocal

Answer 1

x = 1/x - 2

 

Multiply everything by x

x² = x(1/x - 2)

x² = 1 - 2x

x² + 2x - 1 = 0

 

We now have a quadratic equation.  Use the quadratic equation

x = [-b ±√(b²-4ac)]/2a     a=1, b=2, c = -1

 

x = [-2 ±√(2²-4(1)(-1))]/2

x = (-2 ±√8)/2

x = (-2±2√2)/2

x = -1 ±√2

 

Since the problem requires a positive real number and

-1 - √2 would be negative, your answer is:

 

x = -1 + √2

 

check:    x = 1/x -2

-1 + √2 = 1/(-1 + √2 ) -2

-1 + √2 = [1 - 2(-1 + √2)]/(-1 + √2 )

-1 + √2 = (1+2 -2√2)/(-1 + √2 )

(-1 + √2 )² = 3-2√2

1 -2√2  + 2 = 3-2√2

3-2√2 = 3-2√2

 

the equation balances and the answer is correct.