Vieta's Formula 3

Algebra Level 2

If α \alpha and β \beta are the roots of the equation x 2 − m x + 2 = 0 x^2-mx+2=0 and α + 1 β \alpha+\dfrac 1 \beta and β + 1 α \beta+\dfrac 1 \alpha are the roots of the equation x 2 − p x + q = 0 x^2-px+q=0 , what is the value of q q ?


The answer is 4.5.

Junghwan Han by Junghwan Han , 19, South Korea

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1 solution

Junghwan Han
Aug 18, 2019

Due to the Vieta's Formula: α β = 2 \alpha\beta=2 q = ( α + 1 β ) ( β + 1 α ) = α β + α × 1 α + 1 β × β + 1 β α = 2 + 1 + 1 + 1 2 = 9 2 = 4.5 q=\left(\alpha+\frac{1}{\beta}\right)\left(\beta+\frac{1}{\alpha}\right)=\alpha\beta+\alpha\times\frac{1}{\alpha}+\frac{1}{\beta}\times\beta+\frac{1}{\beta\alpha}=2+1+1+\frac{1}{2}=\frac{9}{2}=4.5

0 Helpful 0 Interesting 0 Brilliant 0 Confused

@Junghwan Han , you can just key in \dfrac 1 \alpha 1 α \dfrac 1 \alpha or \cfrac 1 \beta 1 β \cfrac 1 \beta instead of using \displaystyle. { } is not needed after \displaystyle.

Chew-Seong Cheong - 1 year, 9 months ago

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