Determine the value of m - 1/m?

Question

given that 3 = √m + 1/√m

This is a question from a math contest and the answer is supposed to be +/- 3√5. Can anyone provide an algebraic solution?

Answer 1

3 = √m + 1/√m

multiply both sides by √m to give 3√m = m + 1

m - 3√m + 1 = 0

this is a quadratic in √m; so let's rewrite it as u = √m

u^2 - 3u + 1 = 0

we also know that u^2 = m, so the expression for which we're looking for a value is

u^2 - 1/u^2

using the QF, u = [3 +/- sqrt(9 - 4)] / 2

u = [3 +/- sqrt(5)] / 2

let's take the + solution

u = [3 + sqrt(5)] / 2

u^2 = (9 + 6sqrt(5) + 5) / 4

u^2 = (14 + 6sqrt(5)) / 4 = (7 + 3sqrt(5)) / 2

1 / u^2 = 2 / (7 + 3sqrt(5))

rationalize this by multiplying by (7 - 3sqrt(5)) / (7 - 3sqrt(5))

1 / u^2 = 2(7 - 3sqrt(5)) / (49 - 45) = (7 - 3sqrt(5)) / 2

now u^2 - 1/u^2 = m - 1/m = (7 + 3sqrt(5)) / 2 - (7 - 3sqrt(5)) / 2 = 3sqrt(5) / 2 + 3sqrt(5) / 2 = 3sqrt(5)

using the - root will yield -3sqrt(5)

so the two possible answers are m + 1/m = +/- 3sqrt(5)

the key is to recognize that you can make it a quadratic in √m...

Answer 2

3 = m^1/2 + m^-1/2

so 9 = m +1 +1 + 1/m

so 7 = m + 1/m

so 7m = m^2 +1, which means that

m^2 -7m +1 = 0

Use quadratic formula to solve for m and you get m = -7/2 +/- (45)^1/2

give the solution for m, I'll let you carry the ball the rest of the way