**Question: ***Factorise:*

$z^{2}+z+\frac{1}{4}$

**Solution:**
We have:

$z^{2}+z+\frac{1}{4}=z^{2}+2 \times z \times \frac{1}{2} \times\left(\frac{1}{2}\right)^{2}$

$=\left(z+\frac{1}{2}\right)^{2}$

$\therefore z^{2}+z+\frac{1}{4}=\left(z+\frac{1}{2}\right)^{2}$