Sum of Roots = -b/a, Product of Roots =c/a

If the roots of the equation $a{x}^2+2bx+c=0$ be $\text{α}$ and $\text{β}$, and those of the equation $A{x}^2+2Bx+C=0$ be $\text{α}+\text{δ}$ and $\text{β}+\text{δ}$, prove that $\frac{b^2-ac}{a^2}=\frac{B^2-AC}{A^2}$.

If the roots of $l{x}^2+nx+n=0$ be in the ratio of $p:q$, prove that $\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}+\sqrt{\frac{n}{l}}=0$.

If $\text{α}, \text{β}$ are the roots of the equation $x^2-px+q=0$ find the value of

(1) ${\text{α}}^2\left({\text{α}}^2{\text{β}}^{-1}-\text{β}\right)+{\text{β}}^2\left({\text{β}}^2{\text{α}}^{-1}-\text{α}\right)$

(2) $(\text{α}-p{)}^{-4}+(\text{β}-p{)}^{-4}$.

If $x_1$ and $x_2$ are the roots of $a{x}^2+bx+c=0,$ then find the value of ${\left(a{x}_1+b\right)}^{-3}+{\left(a{x}_2+b\right)}^{-3}$.

Discuss the sign of the roots of the equation

$p{x}^2+qx+r=0$

If $\alpha$ and $\beta$ are the roots of $a{x}^2+bx+c=0$, form the equation whose roots are ${\alpha }^2+{\beta }^2$ and ${\alpha }^{-2}+{\beta }^{-2}$.

Find the condition that the one root of $a{x}^2+bx+c=0$ shall be $n$ times of the other.

If $x_1$ and $x_2$ are the roots of $a{x}^2+bx+c=0$, then find the value of ${\left(a{x}_1+b\right)}^{-2}+{\left(a{x}_2+b\right)}^{-2}$.

If $\alpha , \beta$ are the roots of the equation $a{x}^2+bx+c=0$, then find the value of ${\left(\frac{\alpha }{\beta }-\frac{\beta }{\alpha }\right)}^2$.

If $\alpha , \beta$ are the roots of the equation $a{x}^2+bx+c=0,$ then find the value of ${\alpha }^4{\beta }^7+{\alpha }^7{\beta }^4$.

If $\alpha , \beta$ are the roots of the equation $a{x}^2+bx+c=0$, then find the value of $\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$.

For what value of $m$, will the equation $\frac{x^2-bx}{ax-c}=\frac{m-1}{m+1}$ have roots equal in magnitude but opposite in sign?