Question

If roots of the quadratic equation $3a{x}^2+2bx+c=0$ are in the ratio $2:3$, then which of the following statement is true?

• A. $8ac=25b$
• B. $8ac=9b^2$
• C. $8b^2=9ac$
• D. $8b^2=25ac$

Answer 1

The correct option is D $8b^2=25ac$

$3a{x}^2+2bx+c=0letroots:\alpha ,\beta then\left(\frac{\alpha }{\beta }\right)=\left(\frac{2}{3}\right)or\alpha =\left(\frac{2}{3}\right)\beta now\alpha +\beta =\left(\frac{-2b}{3a}\right)\left(\frac{2}{3}\right)\beta +\beta =\left(\frac{-2b}{3a}\right)\left(\frac{5}{3}\right)\beta =\left(\frac{-2b}{3a}\right)or\beta =\left(\frac{-2b}{5a}\right)andthen\alpha \beta =\left(\frac{c}{3a}\right)\left(\frac{2}{3}\right){\beta }^2=\left(\frac{c}{3a}\right)\left(\frac{2}{3}\right)\times (\frac{-2b}{5a}{)}^2=(\frac{c}{3a}\left)\right(\frac{2}{3})\times (\frac{4b^2}{25a^2})=(\frac{c}{3a})8b^2=25ac$