1. The problem asks to find the value(s) of $d$ that complete the square for each quadratic expression. 2. The formula to complete the square for $x^2 + bx + c$ is to find $d = \left(\frac{b}{2}\right)^2$. 3. For each part: (a) Given $x^2 + 5x + d$, here $b=5$. So, $$d = \left(\frac{5}{2}\right)^2 = \frac{25}{4}$$ (b) Given $x^2 - 18x + d$, here $b=-18$. So, $$d = \left(\frac{-18}{2}\right)^2 = (-9)^2 = 81$$ (c) Given $x^2 + dx + 36$, here $c=36$. We want to find $d$ such that $$36 = \left(\frac{d}{2}\right)^2 \implies \left(\frac{d}{2}\right)^2 = 36$$ Taking square root, $$\frac{d}{2} = \pm 6 \implies d = \pm 12$$ (d) Given $x^2 + dx + \frac{49}{4}$, here $c=\frac{49}{4}$. We want $$\frac{49}{4} = \left(\frac{d}{2}\right)^2 \implies \left(\frac{d}{2}\right)^2 = \frac{49}{4}$$ Taking square root, $$\frac{d}{2} = \pm \frac{7}{2} \implies d = \pm 7$$ Final answers: (a) $\frac{25}{4}$ (b) $81$ (c) $12, -12$ (d) $7, -7$