### Quadratic Equations

goals
If the quadratic equation, $ax^2+2cx+b=0$ & $ax^{2}+2bx-5x+c=0 (b \neq c)$ hava a common root then find the value of $a+4b+4c$.
If $x-y=7$ and $xy=9$, find the value of $({x}^{2}+{y}^{2})$.
If $P\left( x \right) ={ ax }^{ 2 }+bx+c$  and $Q\left( x \right) ={ -ax }^{ 2 }+dx+c$, where $ac\neq 0$, then $P\left( x \right) ,Q\left( x \right) =0$
Solve the following inequality:
$\displaystyle \sqrt{x \, + \, 61} \, < \, x \, + \, 5.$
Solve for $x:$
$x^2-5x+6=0$.