Algebra

If one root of quadratic equation $2{ x }^{ 2 }-px+q=0$ is $2+\sqrt { 3}\ (p,q)\in$ rational numbers , then $\dfrac { p }{ q }$ is equal to

For the equation $\displaystyle 3x^{2}+px+3= 0 ; \left ( p> 0 \right ),$ if one of the roots is square of the other then $p$ is equal to

Let $\alpha$ and $\beta$ be the roots of equation $x^{2}-6x-2=0$. If $a_{m}=\alpha^{n}-\beta^{n}$, for $n \ge 1$, then the value of $\dfrac {a_{10}-2a_{8}}{2a_{9}}$ is equal to:

Suppose the quadratic function $f\left( x \right) =a{ x }^{ 2 }+bx+c$ is such that $f\left( -2 \right) =0\quad and\quad \frac { -b }{ 2a } =1$, solve $f\left( x \right)=0$.

If p and q be the roots of the equation $x^2-2x+A=0$ and Let r and s be the roots of the equation $x^2+18x+B=0$. If $p < q < r < s$ are in A.P, then the values of A & B respectively are?

If $\alpha$ and $\beta$ are the roots of the equation $x^{2}-12x+32=0$, then find the value of$\dfrac{{\alpha}^{2}+{\beta}^{2}}{\alpha+\beta}$