Solve the following quadratic equation for $x$.

$x^2+\frac{a}{a+b}x+\frac{a+b}{a}x+1=0$

Solve the following quadratic equation for $x$.

$9x^2-9\left(a+b\right)x+2a^2+5ab+2b^2=0$.

Find the nature of roots of each of the following equations, $x^2+2\sqrt{3}x-9=0$.

Find the nature of roots of each of the following equation, $2x^2+8x+9=0$.

Find the nature of roots of each of the following equations, $3x^2-2\sqrt{6}x+2=0$.

Without solving the following quadratic equation, find the value of $p$ for which the given equation has real and equal roots.

$x^2+(p-3)x+p=0$.