What is the nature of the roots of the equation $a^2x^2 + (a + b)x - b^2 = 0$?

Let $a, b$ and $c$ be the sides of a scalene triangle. If $\lambda$ is a real number such that the roots of the equation $x^2+2(a+b+c)x+3\lambda(ab+bc+ca)=0$ are real,then the interval in which $\lambda$ lies is

If $A$ and $G$ represent the arithmetic mean and geometric mean respectively,and $x^2 - 2Ax + G^2 = 0$,then which of the following is true?

If $P(x) = ax^2 + bx + c$ and $Q(x) = -ax^2 + dx + c$ where $ac \neq 0$, then $P(x) \cdot Q(x) = 0$ has $(a, b, c, d \in \mathbb{R})$:

The polynomial equation of degree $4$ having real coefficients with three of its roots as $2 \pm \sqrt{3}$ and $1+2i$ is:

If $x$ is real,then the minimum value of $x^{2}-8x+17$ is