Question

Find the value of p, for which one of the root of the quadratic equation ${px}^2-14x+8=0is6$ times the other

Answer 1

The given equation is ${px}^2-14x+8=0$

Let the roots be $\alpha ,6\alpha$

Sum of roots is given as $\alpha +6\alpha =\frac{14}{p}$

$\alpha =\frac{2}{p}$

$p=\frac{2}{\alpha }\dots \left(1\right)$

Product of roots is given as $\frac{8}{p}=6{\alpha }^2$

$⟹p=\frac{3}{4{\alpha }^2}$

$⟹\frac{2}{\alpha }=\frac{3}{4{\alpha }^2}$

$⟹2=\frac{3}{4\alpha }$

$⟹\alpha =\frac{3}{8}$

Substituting in $\left(1\right)$

$⟹p=\frac{2}{\frac{3}{8}}$

$⟹p=\frac{16}{3}$