α \alpha and β \beta

Level 2

The two roots of the quadratic equation x 2 + 4 x + 1 = 0 x^2+4x+1=0 are α \alpha and β \beta . If f ( x ) = 0 f(x)=0 is a quadratic equation for x x that has two roots 2 α + β 2\alpha+\beta and α + 2 β \alpha+2\beta , what is the value of f ( 1 ) f(1) ?

23 23 46 46 51 51 33 33

Ajay Dutta by Ajay Dutta , 24, India

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1 solution

Tom Engelsman
Feb 7, 2017

By the Quadratic Formula, the roots are x = − 2 ± 3 x = -2 \pm \sqrt{3} . Set α = − 2 + 3 , β = − 2 − 3 \alpha = -2 + \sqrt{3}, \beta = -2 - \sqrt{3} which produces:

2 α + β = ( − 4 + 2 3 ) + ( − 2 − 3 ) = − 6 + 3 ; 2\alpha + \beta = (-4 + 2\sqrt{3}) + (-2 - \sqrt{3}) = -6 + \sqrt{3};

α + 2 β = ( − 2 + 3 ) + ( − 4 − 2 3 ) = − 6 − 3 . \alpha + 2\beta = (-2 + \sqrt{3}) + (-4 - 2\sqrt{3}) = -6 - \sqrt{3}.

Let f ( x ) = [ x − ( − 6 + 3 ) ] [ x − ( − 6 − 3 ) ] f(x) = [x - (-6 + \sqrt{3})][x - (-6 - \sqrt{3})] so that f ( 1 ) = ( 7 − 3 ) ( 7 + 3 ) = 49 − 3 = 46 . f(1) = (7 - \sqrt{3})(7 + \sqrt{3}) = 49 - 3 = \boxed{46}.

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