The roots of the equation x^2 + ax + b = 0 ar | vedclass

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Question

(A) Given the equation $x^2 + ax + b = 0$ with roots $p$ and $q$. From the relation between roots and coefficients,we have: $p + q = -a$ and $pq = b$.

Vedclass · Accepted Answer

$x^2 + abx + b^3 = 0$

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(A) Given the equation $x^2 + ax + b = 0$ with roots $p$ and $q$. From the relation between roots and coefficients,we have: $p + q = -a$ and $pq = b$. We need to find the equation whose roots are $p^2q$ and $pq^2$. Let the new roots be $\alpha = p^2q$ and $\beta = pq^2$. Sum of the new roots: $\alpha + \beta = p^2q + pq^2 = pq(p + q) = (b)(-a) = -ab$. Product of the new roots: $\alpha \beta = (p^2q)(pq^2) = p^3q^3 = (pq)^3 = b^3$. The required quadratic equation is given by $x^2 - (	ext{sum of roots})x + (	ext{product of roots}) = 0$. Substituting the values: $x^2 - (-ab)x + b^3 = 0$,which simplifies to $x^2 + abx + b^3 = 0$.

The roots of the equation $x^2 + ax + b = 0$ are $p$ and $q$. Then the equation whose roots are $p^2q$ and $pq^2$ will be:

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