1. **State the problem:** Given the quadratic equation $y = x^2 - 7x + 11$, find $a^2 + b^2$ where $a$ and $b$ are the roots. 2. **Recall the formulas:** For a quadratic equation $x^2 + px + q = 0$ with roots $a$ and $b$: - Sum of roots: $a + b = -p$ - Product of roots: $ab = q$ 3. **Identify coefficients:** Here, $p = -7$ and $q = 11$. 4. **Calculate sum and product of roots:** $$a + b = 7$$ $$ab = 11$$ 5. **Use the identity:** $$a^2 + b^2 = (a + b)^2 - 2ab$$ 6. **Substitute values:** $$a^2 + b^2 = 7^2 - 2 \times 11 = 49 - 22 = 27$$ 7. **Final answer:** $$\boxed{27}$$