Answer
$\dfrac{1}{64}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To determine the number that will complete the square to solve the given equation, $
4z^2-z-39=0
,$ use first the properties of equality to express the equation in the form $x^2+bx=c.$ Once in this form, the needed number to complete the square of the left side is equal to $\left( \dfrac{b}{2} \right)^2.$
$\bf{\text{Solution Details:}}$
Using the properties of equality, in the form $x^2+bx=c,$ the given equation is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{4z^2-z-39}{4}=\dfrac{0}{4}
\\\\
z^2-\dfrac{1}{4}z-\dfrac{39}{4}=0
\\\\
z^2-\dfrac{1}{4}z=\dfrac{39}{4}
.\end{array}
In the equation above, $b=
-\dfrac{1}{4}
.$ Using $\left( \dfrac{b}{2} \right)^2$, the number that will complete the square on the left side of the equal sign is
\begin{array}{l}\require{cancel}
\left( \dfrac{-\dfrac{1}{4}}{2} \right)^2
\\\\=
\left( -\dfrac{1}{4}\div2 \right)^2
\\\\=
\left( -\dfrac{1}{4}\cdot\dfrac{1}{2} \right)^2
\\\\=
\left( -\dfrac{1}{8} \right)^2
\\\\=
\dfrac{1}{64}
.\end{array}