If only I can factor more

Algebra Level 3

Suppose that α \alpha and β \beta are the two roots of x 2 + x − 2014 = 0 x^2+x-2014=0 . Then find the value of

α ( α + 2 ) + β ( α + 1 ) \alpha(\alpha+2)+\beta(\alpha+1)

2015 -1 4027 2013

Tom Zhou by Tom Zhou , 21, Canada

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3 solutions

Daniel Liu
Aug 30, 2014

By Vieta's, α + β = − 1 \alpha+\beta = -1 .

Then, α ( α + 2 ) + β ( α + 1 ) = α ( α + β ) + 2 α + β = − α + 2 α + β = α + β = − 1 \begin{aligned}\alpha(\alpha+2)+\beta(\alpha+1)&= \alpha(\alpha+\beta)+2\alpha+\beta\\ &= -\alpha +2\alpha+\beta\\ &= \alpha+\beta\\ &= \boxed{-1}\end{aligned}

5 Helpful 0 Interesting 0 Brilliant 0 Confused

Ah, I was expecting the order of α , β \alpha, \beta to matter. Turns out that it doesn't.

Calvin Lin Staff - 6 years, 9 months ago
Rick B
Sep 5, 2014

α ( α + 2 ) + β ( α + 1 ) = α 2 + 2 α + α β + β = ( α + 1 ) 2 − 1 + β ( α + 1 ) \alpha(\alpha + 2) + \beta(\alpha + 1) = \alpha^2 + 2\alpha + \alpha\beta + \beta = (\alpha + 1)^2 - 1 + \beta(\alpha + 1)

= ( α + 1 ) ( α + 1 + β ) − 1 = (\alpha + 1)(\alpha + 1 + \beta) - 1

By Vieta's, α + β = − 1 \alpha + \beta = -1 , so

( α + 1 ) ( α + 1 + β ) − 1 = ( α + 1 ) ⋅ 0 − 1 = − 1 (\alpha + 1)(\alpha + 1 + \beta) - 1 = (\alpha + 1) \cdot 0 - 1 = \boxed{-1}

2 Helpful 0 Interesting 0 Brilliant 0 Confused
William Isoroku
Sep 18, 2014

Use the fact that a+b=-1.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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