Values of the Symmetric Expression of the Roots

If $\alpha ,\beta$ are the roots of the quadratic equation $a{x}^2+bx+c=0$, then ${\left(\log \left(\frac{1}{\alpha }\right)\right)}^2+{\left(\log \left(\frac{1}{\beta }\right)\right)}^2$  will be the symmetric function of roots.

If $A,B$ are the roots of the quadratic equation $a{x}^2+bx+c=0$, then $\tan (A-B)$  will be the symmetric function of roots.

If $\alpha ,\beta$ are the roots of the quadratic equation $a{x}^2+bx+c=0$, then $\left|\log \left(\frac{\alpha }{\beta }\right)\right|$  will be the symmetric function of roots.

If $\alpha ,\beta$ are the roots of the quadratic equation $a{x}^2+bx+c=0$, then ${\alpha }^2{\beta }^5+{\beta }^2{\alpha }^5$  will be the symmetric function of roots.

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

If $\mathrm{\alpha },\mathrm{ }\mathrm{\beta }$ are the roots of the quadratic equation $\mathrm{a}{\mathrm{x}}^2+\mathrm{b}\mathrm{x}+\mathrm{c}=0$ then the quadratic equation whose roots are ${\mathrm{\alpha }}^3,\mathrm{ }{\mathrm{\beta }}^3$ is

If $\mathrm{\alpha },\mathrm{ }\mathrm{\beta }$ are the roots of $a{x}^2+bx+c=0$ then the quadratic equation whose roots are $2\mathrm{\alpha }+3\mathrm{ }$ and $2\mathrm{\beta }+3$ is

The quadratic equation whose roots are the $\mathrm{x}$ and $\mathrm{y}$ intercepts of the line passing through $\left(1,\mathrm{ }1\right)$ and making a triangle of area $\mathrm{A}$ with the axes may be-

The equation whose roots are reciprocal of the roots of the equation $3{\mathrm{x}}^2-\mathrm{ }20\mathrm{x}+17=0$ is -

If the roots of $a_1{x}^2+b_{1}x+c_1=0$ are ${\alpha }_1, {\beta }_1$ , and those of $a_2{x}^2+b_{2}x+c_2=0$ are ${\alpha }_2, {\beta }_2$ such that ${\alpha }_1{\alpha }_2={\beta }_1{\beta }_2=1$ then-

If $\alpha$ and $\beta$ are roots of the equation ${ax}^2+bx+c=0$, then roots of the equation $a{\left(2x+1\right)}^2- b\left(2x+1\right)\left(3 - x\right)+c{\left(3 - x\right)}^2=0$ are:

If $\alpha ,\beta$ be the roots of the equation $a{x}^2+bx+c=0$, then roots of the equation $a{\left(x+2\right)}^2+b\left(x+2\right)+c=0$ are

If $\alpha \ne \beta$  but ${\alpha }^2=5\alpha -3$  and ${\beta }^2=5\beta -3$  then the equation whose roots are $\frac{\alpha }{\beta }$   and $\frac{\beta }{\alpha }$

Solve  $25^{{\log }_{10}x}=5+4x^{{\log }_{10}5}$.