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Let the quadratic equation \( ax^2 + bx + c = 0 \) have roots \( \alpha \) and \( 2\alpha \). \[ \therefore \alpha + 2\alpha = -\cfrac{b}{a} \] \[ \text{or, } 3\alpha = -\cfrac{b}{a} \] \[ \text{or, } \alpha = -\cfrac{b}{3a} \] Now, \[ \alpha \cdot 2\alpha = \cfrac{c}{a} \] \[ \text{or, } 2\alpha^2 = \cfrac{c}{a} \] \[ \text{or, } 2\times\left(-\cfrac{b}{3a}\right)^2 = \cfrac{c}{a} \] \[ \text{or, } \cfrac{2b^2}{9a^2} = \cfrac{c}{a} \] \[ \text{or, } 2ab^2 = 9a^2c \] \[ \text{or, } 2b^2 = 9ac \quad (\text{Proved}) \]