Nature of Roots

Let $a, b, c$ be the sides of a triangle. No two of them are equal and $\lambda \in R$. If the roots of the equation $x^2+2\left(a+b+c\right)x+3\lambda \left(ab+bc+ca\right)=0$ are real, then

The smallest value of $k$, for which both the roots of the equation, $x^2-8kx+16\left(k^2-k+1\right)=0$ are real, distinct and have value atleast $4$, is

If the roots of the quadratic equation $x^2+8x+{\mu }^2+6\mu =0$ are real, then $\mu$ lies between

Find all the integral values of $a$ for which the quadratic equation $\left(x-a\right)\left(x-10\right)+1=0$ has integral roots.

Prove that if the roots of the equation $\left(a^4+b^4\right)x^2+4abcdx+\left(c^4+d^4\right)=0$ be real, then they cannot be unequal.

If the roots of the equation $x^2-2\left(a+b\right)x+a\left(a+2b+c\right)=0$ be equal, prove that $a,b,c$ are in geometric progression.

If the roots of the equation $a\left(b-c\right)x^2+b\left(c-a\right)x+c\left(a-b\right)=0$ are equal, then prove that $\frac{1}{a}+\frac{1}{c}=\frac{2}{b}$.

If the roots of the equation $q{x}^2-px+q=0$ be a real show that $p$ cannot lie between $-2q$ and $2q\left(q>0\right)$.

If the roots of the equation $q{x}^2+2px+2q=0$ are real and unequal, prove that the roots of the equation $\left(p+q\right)x^2+2qx+\left(p-q\right)=0$ are imaginary.

If the roots of the equations $q{x}^2+px+q=0$ be imaginary, show that the roots of the equation $x^2-4qx+p^2=0$ are real and unequal.

If $a, b, c$ are three rational numbers and if $\sqrt{2}$ is a root of the quadratic equation $a{x}^2+bx+c=0$, what is the vlaue of $b$?

Verify that the discriminant of the quadratic equation $x^2-2\sqrt{5}x+1=0$, is a perfect square, yet its roots are not conjugate surds. Explain the reason behind.

Verify that for the equation $x^2-5ix-6=0$, the roots are not conjugate imaginary numbers. Explain the reason

Find the nature of roots of the following equations without determining the roots :

$\frac{3}{2}i{x}^2+4x+\frac{3i}{2}=0$

Find the nature of roots of the following equations without determining the roots :

$\frac{7}{8}{x}^2-x+\frac{2}{7}=0$

Find the nature of roots of the following equations without determining the roots :

$2x^2-2\sqrt{6}x+3=0$