1. **Problem statement:** Find the possible values of $b$ such that the expressions are perfect squares. 2. **Recall:** A quadratic expression $ax^2 + bx + c$ is a perfect square if it can be written as $(x + d)^2 = x^2 + 2dx + d^2$ for some $d$. 3. **For (a) $x^2 + bx + 1$:** - Assume it is a perfect square: $(x + d)^2 = x^2 + 2dx + d^2$ - Equate coefficients: $b = 2d$ and $1 = d^2$ - Solve $d^2 = 1$ gives $d = \\pm 1$ - Then $b = 2d = 2$ or $-2$ 4. **For (c) $x^2 + 2bx + 81$:** - Assume it is a perfect square: $(x + d)^2 = x^2 + 2dx + d^2$ - Equate coefficients: $2b = 2d$ and $81 = d^2$ - From $81 = d^2$, $d = \\pm 9$ - Then $2b = 2d$ implies $b = d = 9$ or $-9$ **Final answers:** - (a) $b = 2$ or $b = -2$ - (c) $b = 9$ or $b = -9$