If $a+b=2$ and $a^4+b^4=272$ then a quadratic equation whose roots are a and b is

We have,

$a+b=2$ and $a^4+b^4=272$ 

Now, $a^4+b^4=272$

$\begin{array}{l}\Rightarrow {\left(a^2+{\dot{b}}^2\right)}^2-2a^2{b}^2=272\\ \Rightarrow {\left\{(a+b{)}^2-2ab\right\}}^2-2a^2{b}^2=272\\ \Rightarrow (4-2ab{)}^2-2a^2{b}^2=272\\ \Rightarrow 16-16ab+2a^2{b}^2=272\\ \Rightarrow (ab{)}^2-8(ab)-128=0\\ \Rightarrow (ab-16)(ab+8)=0\Rightarrow ab=16\text{ or },-8\end{array}$

Thus, we have $a+b = 2$ and $ab=16$, or -8.

So, the quadratic equations whose roots are a and b are 

$x^2-2x+16=0\text{ and }{x}^2-2x-8=0$