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Question

(B) Let $\alpha$ and $\beta$ be the roots of $ax^2 + bx + c = 0$.Then $\alpha + \beta = -b/a$ and $\alpha \beta = c/a$.The new roots are $\alpha - 1$

Vedclass · Accepted Answer

$b = -c$

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(B) Let $\alpha$ and $\beta$ be the roots of $ax^2 + bx + c = 0$.Then $\alpha + \beta = -b/a$ and $\alpha \beta = c/a$.The new roots are $\alpha - 1$ and $\beta - 1$.The new equation is $2x^2 + 8x + 2 = 0$,which can be written as $x^2 + 4x + 1 = 0$.Sum of new roots: $(\alpha - 1) + (\beta - 1) = -4$ $\Rightarrow \alpha + \beta - 2 = -4$ $\Rightarrow \alpha + \beta = -2$.Since $\alpha + \beta = -b/a$,we have $-b/a = -2 \Rightarrow b = 2a$.Product of new roots: $(\alpha - 1)(\beta - 1) = 1 \Rightarrow \alpha \beta - (\alpha + \beta) + 1 = 1$.Substituting the values: $c/a - (-2) + 1 = 1$ $\Rightarrow c/a + 3 = 1$ $\Rightarrow c/a = -2$.Since $c/a = -2$ and $b/a = 2$,we have $c/a = -b/a$,which implies $c = -b$ or $b = -c$.

The equation formed by decreasing each root of $ax^2 + bx + c = 0$ by $1$ is $2x^2 + 8x + 2 = 0$. Then:

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