problem: Let the roots of the polynomial $x^2 + 7x - 2$ be $\alpha$ and $\beta.$ Evaluate $\alpha^2 + \beta^2.$
solution: We make use of the following fact:
  
  "For a quadratic equation $ax^2 + bx + c,$ the sum of the roots is $-b/a$ while the product of the roots is $c/a.$'' Hence, $\alpha + \beta = -7$ and $\alpha*\beta = -2.$
  
  Now, we use the fact that $(\alpha + \beta)^2 - 2\alpha\beta =\alpha^2 + \beta^2.$ Or, $7^2 + 4 = \alpha^2 + \beta^2.$
  
  Thus the answer is $\boxed{53}.$