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

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Solution

Let the roots of equation of $a x^{2} + b x + c$ be $p, q$
The roots of $2 x^{2} + 8 x + 2 = 0$ is $p - 1, q - 1$
Sum of roots $p - 1 + q - 1 = \frac{8}{2} p + q = 6$
Product of roota $(p - 1) (q - 1) = \frac{2}{2} p q - (p + q) - 1 = 1 p q = 8$
$⟹ a x^{2} + b x + c = x^{2} - 6 x + 8 = 0 ⟹ a = 1, b = - 6, c = 8 \frac{a}{b} = - \frac{1}{6}, \frac{b}{c} = \frac{- 3}{4}$

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