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

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(A) Given the quadratic equation $x^2 + ax + b = 0$ with roots $p$ and $q$.From the properties of roots,we have the sum of roots $p + q = -a$ and the

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

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(A) Given the quadratic equation $x^2 + ax + b = 0$ with roots $p$ and $q$.From the properties of roots,we have the sum of roots $p + q = -a$ and the product of roots $pq = b$.We need to find the equation whose roots are $\alpha = p^2q$ and $\beta = pq^2$.The sum of the new roots is $\alpha + \beta = p^2q + pq^2 = pq(p + q)$.Substituting the known values,we get $\alpha + \beta = (b)(-a) = -ab$.The product of the new roots is $\alpha \cdot \beta = (p^2q)(pq^2) = p^3q^3 = (pq)^3$.Substituting the known value,we get $\alpha \cdot \beta = b^3$.The required quadratic equation is given by $x^2 - (	ext{sum of roots})x + (	ext{product of roots}) = 0$.Substituting the values,we get $x^2 - (-ab)x + b^3 = 0$,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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