If 3p^2 = 5p + 2 and 3q^2 = 5q + 2 where p ne | vedclass

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Question

(A) Given that $p$ and $q$ are roots of the quadratic equation $3x^2 - 5x - 2 = 0$.
From the properties of roots,the sum of roots is $p + q = -(-5)/3

Vedclass · Accepted Answer

$3x^2 - 5x - 100 = 0$

Vedclass · Answer

(A) Given that $p$ and $q$ are roots of the quadratic equation $3x^2 - 5x - 2 = 0$.
From the properties of roots,the sum of roots is $p + q = -(-5)/3 = 5/3$ and the product of roots is $pq = -2/3$.
We need to find the equation whose roots are $\alpha = 3p - 2q$ and $\beta = 3q - 2p$.
Sum of roots $(\alpha + \beta) = (3p - 2q) + (3q - 2p) = p + q = 5/3$.
Product of roots $(\alpha \beta) = (3p - 2q)(3q - 2p) = 9pq - 6p^2 - 6q^2 + 4pq = 13pq - 6(p^2 + q^2)$.
Since $p^2 + q^2 = (p + q)^2 - 2pq = (5/3)^2 - 2(-2/3) = 25/9 + 4/3 = 37/9$.
Product of roots $= 13(-2/3) - 6(37/9) = -26/3 - 74/3 = -100/3$.
The required quadratic equation is $x^2 - (	ext{sum of roots})x + (	ext{product of roots}) = 0$.
$x^2 - (5/3)x - 100/3 = 0$,which simplifies to $3x^2 - 5x - 100 = 0$.

If $3p^2 = 5p + 2$ and $3q^2 = 5q + 2$ where $p \ne q$,then the equation whose roots are $3p - 2q$ and $3q - 2p$ is

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