The equation formed by decreasing the roots of the quadractic equation $a x^{2} + b x + c = 0$ by $1$ is $2 x^{2} + 8 x + 2 = 0$ , then which of the following statement(s) is/are true?

a : b : c = 1 : 2 : - 2

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a : b : c = 2 : 1 : - 2

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b = - c

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a = - c

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Solution

The correct options are
A $a : b : c = 1 : 2 : - 2$
C $b = - c$
Let the roots of the equation $f (x) = a x^{2} + b x + c = 0$ be $α, β$
Now, the equation with roots $α - 1, β - 1$ is,
$f (x + 1) = 0 \Rightarrow a (x + 1)^{2} + b (x + 1) + c = 0$
$\Rightarrow a x^{2} + (2 a + b) x + (a + b + c) = 0$
Comparing to the equation
$2 x^{2} + 8 x + 2 = 0$
So,
$\frac{a}{2} = \frac{(2 a + b)}{8} = \frac{a + b + c}{2} = k$
Where is some constant,
$\Rightarrow a = 2 k \Rightarrow 2 a + b = 8 k \Rightarrow b = 4 k \Rightarrow a + b + c = 2 k \Rightarrow c = - 4 k$

Therefore,
$b = - c, b = 2 a, c = - 2 a$
$a : b : c = 1 : 2 : - 2$

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