Quadratic Equations

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 $$x:$$
$$x^2-5x+6=0$$.