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Add Square Roots, Get Real(m) (Posted on 2007-05-06) Difficulty: 3 of 5
Determine all real m satisfying this equation:

√(m - 1/m) + √(1 - 1/m) = m

.

See The Solution Submitted by K Sengupta    
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Solution The unique solution Comment 3 of 3 |

For the equation to have a solution over R, m>=1 from the constraints placed by the expressions under the radicals.

Rearrange the equation as follows:
sqrt(m - 1/m) = m - sqrt(1 - 1/m)

Square both sides and simplify:
m^2 - m + 1 = 2m*sqrt(1 - 1/m)

Square again and simplify:
m^4 - 2m^3 - m^2 + 2m + 1 = 0

The polynomial factors as:
(m^2 - m - 1)^2 = 0

The two roots of the polynomial are:
(1+sqrt(5))/2 and (1-sqrt(5))/2

Since only one root satisfies the constraint m>=1, there is a unique solution of m=(1+sqrt(5))/2


  Posted by Brian Smith on 2007-05-06 23:06:05
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