Theory question 102
PolynomialsIf α and β are the roots of the equation 2x2 − 7x + 4 = 0, find the equation whose roots are α/β and β/α.
Show the model answer
Model answer
α + β = 7/2, αβ = 2
Sum of new roots = (α2 + β2)/αβ = [(α + β)2 − 2αβ]/αβ = (49/4 − 4)/2 = 33/8
Product of new roots = (α/β)(β/α) = 1
Equation: x2 − (33/8)x + 1 = 0, i.e. 8x2 − 33x + 8 = 0
Explanation
Write α² + β² as (α + β)² − 2αβ.