If alpha and beta are the roots of ax^2 + bx | vedclass

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(D) Given that $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$.From the properties of quadratic equations,we have $\alpha + \beta = -b/a$ an

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$ax^2 + x(b - 4a) + 4a - 2b + c = 0$

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(D) Given that $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$.From the properties of quadratic equations,we have $\alpha + \beta = -b/a$ and $\alpha\beta = c/a$.Let the new roots be $\alpha' = 2 + \alpha$ and $\beta' = 2 + \beta$.The sum of the new roots is $\alpha' + \beta' = (2 + \alpha) + (2 + \beta) = 4 + (\alpha + \beta) = 4 - b/a = (4a - b)/a$.The product of the new roots is $\alpha'\beta' = (2 + \alpha)(2 + \beta) = 4 + 2(\alpha + \beta) + \alpha\beta = 4 + 2(-b/a) + c/a = (4a - 2b + c)/a$.The required quadratic equation is given by $x^2 - (	ext{sum of roots})x + (	ext{product of roots}) = 0$.Substituting the values: $x^2 - ((4a - b)/a)x + (4a - 2b + c)/a = 0$.Multiplying the entire equation by $a$,we get $ax^2 - (4a - b)x + (4a - 2b + c) = 0$.This simplifies to $ax^2 + x(b - 4a) + 4a - 2b + c = 0$.

If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$,then the equation whose roots are $2 + \alpha$ and $2 + \beta$ is:

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