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α and β are the roots of $x^{2} - 5 x + 3 = 0$ . Which equation has roots α + 1/β and β + 1/α?

Competition practiceAlgebra

A $3 x^{2} - 20 x + 16 = 0$
B $3 x^{2} - 16 x + 20 = 0$
C $x^{2} - 20 x + 16 = 0$
D $3 x^{2} + 20 x + 16 = 0$

Step-by-Step Solution

Sum = (α + β) + (α + β)/αβ = 5 + 5/3 = 20/3. Product =

(α β + 1)^{2}

/αβ = 16/3. So

x^{2} - (20/3) x + 16/3 = 0

, i.e.

3 x^{2} - 20 x + 16 = 0

.

Answer: A)

3 x^{2} - 20 x + 16 = 0

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