Find the values!

Algebra Level 3

Given that α , γ \alpha ,\gamma are the roots of the equation A x 2 − 4 x + 1 = 0 A{ x }^{ 2 }-4x+1=0 and β , δ \beta ,\quad \delta the roots of the equation, B x 2 − 6 x + 1 = 0 B{ x }^{ 2 }-6x+1=0 , find the value of A and B, such that α , β , γ , δ \alpha ,\beta ,\gamma ,\delta are in HP

NOTE:- HP means Harmonic Progression

A=6, B=5 A=5, B=6 A=3, B=8 A=8, B=3

Manish Dash by Manish Dash , 22, India

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2 solutions

Harry Stuart
Jul 7, 2016

I know this is a cheating way, but using the discriminant in the quadratic formula, b²-4ac > 0 must be true for the quadratic to have two real solutions. In Ax²-4x+1 then, 16-4A > 0 and A < 4. Therefore A=3, B=8 is the only possible solution from the solutions given

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Ramiel To-ong
Jun 8, 2015

Using Vieta's formula: A = 3 and B = 8 the roots will be in HP. 1/1, 1/2,1/3 and 1/4

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