EXERCISE 6.4

Two candidates attempt to solve a quadratic equation of the form $x^2+px+q=0$. One starts with a wrong value of $p$ and obtains the roots as $2$ and $6$. The other starts with a wrong value of $q$ and obtains the roots as $2$ and $-9$. Find the correct roots and the quadratic equation.

If $\alpha ,\beta$ are the roots of the equation $x^2-7x-4=0$, then show that ${\alpha }^3=53\alpha +28$

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

Form an equation whose roots are $3,\frac{-1}{3}$

Form an equation whose roots are $\sqrt{3},\frac{1}{\sqrt{3}}$

Form an equation whose roots are $0,0$

Form an equation with rational coefficients one of whose roots is $\frac{\sqrt{3}+1}{\sqrt{3}-1}$

Form an equation with real coefficients one of whose roots is $-1-\sqrt{-4}$