The roots of the given equation $2(a^2 + b^2)x^2 + 2(a + b)x + 1 = 0$ are

If roots of the equation $ax^2 + bx + c = 0$ are $(\alpha - \beta)$ and $(\gamma - \delta)$,and roots of the equation $Ax^2 + Bx + C = 0$ are $(\alpha + \delta)$ and $(\beta + \gamma)$,then $\left| \frac{a}{A} \right|$ is equal to (where $D_1$ and $D_2$ are discriminants of the given equations respectively).

Let $S = \{ \sin^2 2\theta : (\sin^4 \theta + \cos^4 \theta)x^2 + (\sin 2\theta)x + (\sin^6 \theta + \cos^6 \theta) = 0 \text{ has real roots} \}$. If $\alpha$ and $\beta$ are the smallest and largest elements of the set $S$,respectively,then $3((\alpha - 2)^2 + (\beta - 1)^2)$ equals:

The number of ordered pairs $(a, b)$ of integers such that $1 \leq a, b \leq 2021$ and the equations $x^2 - ax + b = 0$ and $x^3 - ax^2 + bx + a - b = 0$ have a common real root is

The quadratic equation whose roots are the numbers having arithmetic mean $34$ and geometric mean $16$ is

If the roots of the equation $x^2 - 8x + (a^2 - 6a) = 0$ are real,then: