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Q.
Find the condition to be satisfied by the coefficients of the equation $p x^2+q x+r=0$, so that the roots are in the ratio $3: 4$.
Quadratic Equations
Solution:
Let the common factor for the roots be $\gamma$.
The roots are $3 y$ and $4 y, 3 y+4 y=\frac{-q}{p}$ $\Rightarrow y=\frac{-q}{7 p}$
$(3 y)(4 y)=\frac{r}{p} \Rightarrow y^2=\frac{r}{12 p}$
$\gamma^2=\left(\frac{-q}{7 p}\right)^2=\frac{r}{12 p} 12 q^2=49 p r$