Given, x2+x−2=0x^{2} + x - 2 = 0x2+x−2=0, a = 1, b = 1 and c = -2.
α+β= −ba= −11=−1\alpha + \beta = \;\frac{-b}{a} = \;\frac{-1}{1} = -1α+β=a−b=1−1=−1
αβ= ca= −21=−2\alpha\beta = \;\frac{c}{a} = \;\frac{-2}{1} = -2αβ=ac=1−2=−2
1α2+ 1β2= β2 + α2(αβ)2\;\frac{1}{\alpha^{2}} + \;\frac{1}{\beta^{2}} = \;\frac{\beta^{2}\;+\;\alpha^{2}}{(\alpha\beta)^{2}}α21+β21=(αβ)2β2+α2
β2+α2=(α+β)2−2αβ=(−1)2−2(−2)=1+4=5\beta^{2} + \alpha^{2} = (\alpha + \beta)^{2} - 2\alpha\beta = (-1)^{2} - 2(-2) = 1 + 4 = 5β2+α2=(α+β)2−2αβ=(−1)2−2(−2)=1+4=5
1α2+ 1β2= 5(−2)2= 54\;\frac{1}{\alpha^{2}} + \;\frac{1}{\beta^{2}} = \;\frac{5}{(-2)^{2}} = \;\frac{5}{4}α21+β21=(−2)25=45.