Nature of the Roots of a Quadratic Equation

If $a$ and $b$ are real and $a\ne b$ then show that the roots of the equation $(a-b)x^2+5(a+b)x-2(a-b)=0$ are real and unequal.

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

Find the values of $k$ for which the given quadratic equation has real and distinct root:
$5x^2-kx+1=0$

Find the values of $k$ for which the given quadratic equation has real and distinct roots:

$x^2-kx+9=0$

Find the values of $k$ for which the given quadratic equation has real and distinct roots:
$9x^2+3kx+4=0$

Find the value of $p$ for which the quadratic equation $(2p+1)x^2-(7p+2)x+(7p-3)=0$ has real and equal roots.

Find the values of $k$ for which the given quadratic equation has real and distinct roots:
$k{x}^2+6x+1=0$

For what values of $p$ are the roots of the equation $4x^2+px+3=0$ real and equal?

Find the value of $k$ for which the roots of $9x^2+8kx+16=0$ are real and equal

If the roots of the quadratic equation $\left(c^2-ab\right)x^2-2\left(a^2-bc\right)x+\left(b^2-ac\right)=0$ are real and equal, show that either $a=0$ or $\left(a^3+b^3+c^3=3abc\right)$

If $a$ and $b$ are distinct real numbers, show that the quadratic equations
$2(a^2+b^2)x^2+2(a+b)x+1=0$ has no real roots.

Find the value of $p$ for which the quadratic equation $2x^2+px+8=0$ has real roots.

Find the value of $\alpha$ for which the equation $(\alpha -12)x^2+2(\alpha -12)x+2=0$ has equal roots.

Find the values of $p$ for which the quadratic equation $(p+1)x^2-6(p+1)x+3(p+9)=0$, $p\ne -1$ has equal roots. Hence find the roots of the equation.

If $-4$ is a root of the equation $x^2+2x+4p=0$ find the value of $k$ for the which the quadratic equation $x^2+px(1+3k)+7(3+2k)=0$ has equal roots.

Find the values of $k$ for which the quadratic equation $(3k+1)x^2+2(k+1)x+1=0$ has real  and equal roots.

Find the nature of roots of the following quadratic equations:
$x^2-x+2=0$

If $3$ is a root of the quadratic equation $x^2-x+k=0$ find the value of $p$ so that the roots of the equation $x^2+k(2x+k+2)+p=0$ are equal.

If $-5$ is a root of the quadratic equation $2x^2+px-15=0$ and the quadratic equation $p\left(x^2+x\right)+k=0$ has equal roots, find the value of $k$.

Show that the roots of the equation $x^2+px-q^2=0$ are real for all real values of $p$ and $q$