Solve the equation (x+1/x)^2 - 5(x+1/x) + 6 = 0 when m = (x+1/x)

Question

Please show working out. Ty.

Answer 1

 

The equationis  (x+1/x)^2 - 5(x+1/x) + 6 = 0

m = x + 1/ x, so the equation is m^2 - 5m + 6 = 0

m^2 - 2m - 3m + 6 = 0

m(m - 2) - 3 (m -2 ) = 0

( m - 2)(m - 3)

Apply zero product property.

m -2 = 0 or m - 3 = 0

m = 2 or m = 3.

x + 1/ x = 2 or 3.

 

Solve for case1

x + 1/ x = 2

x^2 + 1 = 2x

x^2  - 2x + 1 = 0

(x - 1)^2 = 0

x = 1.

Solve for case2

x + 1/ x = 3

x^2 + 1 = 3x

x^2  - 3x + 1 = 0

Apply quadratic formula

x = 1/2 (3 + sqrt5), x = 1/2 (3 -sqrt5)