Question

If \( \alpha, \beta \) are the roots of the equation \( x^{2}-5 x+4=0 \). find the equation whose roots \( \operatorname{are} \alpha+\frac{1}{\beta}, \beta+\frac{1}{\alpha} \)

Answer 1

alpha can equal 4 and beta can equal 1 (or vice versa), so

the equation satisfying the conditions is:

x² + ²⁵/₄ x + ²⁵/₄

Answer 2

Given α + β = 5 and αβ = 4

the new roots α + 1/β,   β +1/α

sum of roots = α + 1/β + β +1/α = α + β + ( α + β)/(αβ)  = 5 + 5/4 = (25)/4

product of roots = (α + 1/β )( β +1/α ) = αβ + 1 + 1 + 1/(αβ) = 4 + 1 + 1 + 1/4 = (25)/4

Equation with new roots

x² - ( sum of roots ) x + ( product of roots ) = 0

x² - ( (25)/4 ) x + (25)/4 = 0

4x² - 25x + 25 = 0