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Question Number 56904 by pete last updated on 26/Mar/19
If α and β are the roots of of the equation  3x^2 −x−3=0, find thevalue of (α^2 −β^2 )  if α>β.
$$\mathrm{If}\:\alpha\:\mathrm{and}\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{3x}^{\mathrm{2}} −\mathrm{x}−\mathrm{3}=\mathrm{0},\:\mathrm{find}\:\mathrm{thevalue}\:\mathrm{of}\:\left(\alpha^{\mathrm{2}} −\beta^{\mathrm{2}} \right) \\ $$$$\mathrm{if}\:\alpha>\beta. \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 26/Mar/19
(α−β)^2 =(α+β)^2 −4αβ  α+β=((−b)/a)=((−(−1))/3)=(1/3)  αβ=(c/a)=((−3)/3)=−1                    (α−β)^2 =(α+β)^2 −4αβ                                     =(1/9)+4=((37)/9)  (α−β)=((√(37))/3)  α^2 −β^2 =(α+β)(α−β)                =(1/3)×((√(37))/3)=((√(37))/9)
$$\left(\alpha−\beta\right)^{\mathrm{2}} =\left(\alpha+\beta\right)^{\mathrm{2}} −\mathrm{4}\alpha\beta \\ $$$$\alpha+\beta=\frac{−{b}}{{a}}=\frac{−\left(−\mathrm{1}\right)}{\mathrm{3}}=\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\alpha\beta=\frac{{c}}{{a}}=\frac{−\mathrm{3}}{\mathrm{3}}=−\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\alpha−\beta\right)^{\mathrm{2}} =\left(\alpha+\beta\right)^{\mathrm{2}} −\mathrm{4}\alpha\beta \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{1}}{\mathrm{9}}+\mathrm{4}=\frac{\mathrm{37}}{\mathrm{9}} \\ $$$$\left(\alpha−\beta\right)=\frac{\sqrt{\mathrm{37}}}{\mathrm{3}} \\ $$$$\alpha^{\mathrm{2}} −\beta^{\mathrm{2}} =\left(\alpha+\beta\right)\left(\alpha−\beta\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{1}}{\mathrm{3}}×\frac{\sqrt{\mathrm{37}}}{\mathrm{3}}=\frac{\sqrt{\mathrm{37}}}{\mathrm{9}} \\ $$
Commented by pete last updated on 26/Mar/19
Thanks for your help.
$$\mathrm{Thanks}\:\mathrm{for}\:\mathrm{your}\:\mathrm{help}. \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 26/Mar/19
most welcome...
$${most}\:{welcome}… \\ $$

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