Do not find the roots - Re-uploaded

Algebra Level 3

Given that the roots of the equation 2 x 2 + 4 x − 5 2x^2 + 4x -5 are α \alpha and β \beta , find

α ( 1 − β ) + β α 2 + β 2 \frac{\alpha (1-\beta) + \beta}{\alpha ^2 + \beta ^2}

Re-uploaded due to mix up


The answer is 0.05555555555.

James Watson by James Watson , 16, United Kingdom

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1 solution

Chew-Seong Cheong
Aug 22, 2020

By Vieta's formula , we have α + β = − 4 2 = 2 \alpha +\beta=-\dfrac 42=2 and α β = − 5 2 \alpha \beta = -\dfrac 52 . Then

α ( 1 − β ) + β α 2 + β 2 = α + β − α β ( α + β ) 2 − 2 α β = − 2 + 5 2 4 + 5 = 1 2 9 = 1 18 ≈ 0.0556 \begin{aligned} \frac {\alpha (1 - \beta) +\beta} {\alpha ^2 +\beta ^2} & = \frac {\alpha +\beta - \alpha \beta} {(\alpha +\beta) ^2 - 2\alpha \beta} \\ & = \frac {-2+\frac 52}{4+5} \\ & = \frac {\frac 12}9 = \frac 1{18} \approx \boxed {0.0556} \end{aligned}

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