Find the value of c that would make the expression a perfect square. z^2+z+c

To make the expression $z^2 + z + c$ a perfect square, we need to find a value of $c$ such that the expression can be written in the form $(z+a)^2$ for some value $a$.<br /><br />Expanding $(z+a)^2$, we get:<br />$(z+a)^2 = z^2 + 2az + a^2$<br /><br />Comparing this with the given expression $z^2 + z + c$, we can see that the coefficient of $z$ is 1, so we must have $2a = 1$. This means $a = \frac{1}{2}$.<br /><br />Now, we need to find the value of $c$. Since $(z+a)^2 = z^2 + 2az + a^2$, we have $c = a^2$. Substituting $a = \frac{1}{2}$, we get:<br />$c = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$<br /><br />Therefore, the value of $c$ that makes the expression a perfect square is $\frac{1}{4}$.<br /><br />We can check this:<br />$z^2 + z + \frac{1}{4} = \left(z + \frac{1}{2}\right)^2$<br /><br />Final Answer: The final answer is $\boxed{1/4}$<br />