Question

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

• A. $3x^2-5x-100=0$
• B. $5X^2+3x+100=0$
• C. $3x^2-5x+100=0$
• D. $5x^2-3x-100=0$

Answer 1

The correct option is A $3x^2-5x-100=0$

Let $f\left(x\right)=3x^{-}5x-2$ roots $p,qp+q=5/3pq=-2/3$

Equation whose roots are $3p-2q$ & $3q-2p$

$x^2-(3p-2q+3q-2p)x+(3p-2q)(3q-2p)=0$

$x^2-(p+q)x+(9pq-6p^2-6p^2-6q^2+4pq)=0$

$x^2-(p+q)x+(13pq-6(p^2+q^2\left)\right)=0$

$x^2-(p+q)x+(13pq\ast 6((p^2+q^2)-2pq\left)\right)=0$

$x^2-(p+q)x+(13pq-6(p+a{)}^2+12pq)=0$

$x^2-(p+a)x+(25pq-6(p+q{)}^2)=0$

$x^2-\frac{5x}{3}+\left(25(\frac{-2}{3}-6{\left(\frac{5}{3}\right)}^2)\right)=0$

$x^2-\frac{5x}{3}+(-\frac{50}{3}-\frac{150}{9})=0$

$x^2-\frac{5x}{3}+(-\frac{50}{3}-\frac{50}{3})=0$

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