If the equation formed by decreasing each root of the `a x^2+b x+c=0` by 1 `2x^2+8x+2=0` . Find the condition.

Answer

`(a)/(1)=(b)/(2)=(c)/(-2)`

Explanation

Let ` alpha , beta ` the roots of the given equation `ax^(2) + bx + c = 0`
Then , `alpha + beta = - (b)/(a), alpha beta = (c)/(a)`
Now roots of equation ` 2x^(2) + 8x + 2 = 0 are alpha - 1, beta - 1 `. Their sum is
` alpha + beta - 2 = - (b)/(a) - 2 = - (8)/(2) = -4`
`rArr (b)/(a) = 2`
Their product is
` (alpha - 1) (beta - 1) = alpha beta - (alpha + beta) + 1`
= ` (c)/(a) + (b)/(a) + 1 = 1`
`because ` New equation is `2x^(2) + 8x + 2 = 0`
or c + b = 0
or b = -c
`rArr (a)/(1) = (b)/(2) = (c)/(-2)` .