αβ+ βα=α2+β2αβ\;\frac{\alpha}{\beta} + \;\frac{\beta}{\alpha} = \frac{\alpha^{2} + \beta^{2}}{\alpha\beta}βα+αβ=αβα2+β2
α+β=−ba\alpha + \beta = \frac{-b}{a}α+β=a−b; αβ=ca\alpha\beta = \frac{c}{a}αβ=ac
α2+β2=(α+β)2−2(αβ)\alpha^{2} + \beta^{2} = (\alpha + \beta)^{2} - 2(\alpha\beta)α2+β2=(α+β)2−2(αβ)
From the equation, a = 2, b = -3, c =4
α+β=−(−3)2=32\alpha + \beta = \frac{-(-3)}{2} = \frac{3}{2}α+β=2−(−3)=23
αβ=42=2\alpha\beta = \frac{4}{2} = 2αβ=24=2
α2+β2=(32)2−2(2)=94−4=−74\alpha^{2} + \beta^{2} = \left(\frac{3}{2}\right)^{2} - 2(2) = \frac{9}{4} - 4 = \frac{-7}{4}α2+β2=(23)2−2(2)=49−4=4−7
⟹ αβ+βα=−742=−78\implies \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\frac{-7}{4}}{2} = \frac{-7}{8}⟹βα+αβ=24−7=8−7