Question

If the equation formed by decreasing each root of $a{x}^2+bx+c=0$ by $1$ is $2x^2+8x+2=0$, then find the values of $\frac{a}{b}$ and $\frac{b}{c}$.

Answer 1

Let the roots of equation of $a{x}^2+bx+c$ be $p,q$

The roots of $2x^2+8x+2=0$ is $p-1,q-1$

Sum of roots $p-1+q-1=\frac{8}{2}p+q=6$

Product of roota $(p-1)(q-1)=\frac{2}{2}pq-(p+q)-1=1pq=8$

$⟹a{x}^2+bx+c=x^2-6x+8=0⟹a=1,b=-6,c=8\frac{a}{b}=-\frac{1}{6},\frac{b}{c}=\frac{-3}{4}$