If the roots of the equation $a x^{2} + b x + c = 0$ bear to one another the ratio $3 : 4$ . Prove that $12 b^{2} = 49 a c$

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Solution

Let they be $3 x, 4 x$ ;
$7 x = \frac{- b}{a}; 12 x^{2} = \frac{c}{a} ⟹ x^{2} = \frac{c}{12 a} = \frac{b^{2}}{49 a^{2}} ⟹ 12 b^{2} = 49 a c$

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