Basic Quadratics

According to the question

$\dfrac{x+y}{2}=7\implies{(1)}$ $\text{and}$ $\sqrt{xy}=\sqrt{19}\implies{(2)}$

Squaring (2) we have $\color{#D61F06}{xy=19}$

From (1) we have

$x^2+y^2+2\color{#D61F06}{xy} = 196\implies x^2+y^2=196-38=\boxed{158}$

Given: $\frac{x+y}{2}=7\Longrightarrow x+y=14;$ $\sqrt{xy}=\sqrt{19}\Longrightarrow xy=19.$ Now $x^2+y^2=(x+y)^2-2xy=14^2-2\cdot 19=196-38=158.$