169/4
1) Examining the quadratic x² -13x+ 31 = 0 and knowing that:
a= 1
b = -13
c =31
2) Then we can write the following number to add the next square
x² -13x+ 31 Pick that b coefficient and then divide it by 2
[tex]\begin{gathered} x^2-13x+31=0 \\ x^2-13x\text{ =-31} \\ (x^{}-\frac{13}{2})^2=-31+\frac{169}{4} \\ (x^{}-\frac{13}{2})^2=\frac{45}{4} \\ \sqrt[]{(x^{}-\frac{13}{2})^2}=\sqrt[]{\frac{45}{4}} \\ x-\frac{13}{2}=\pm\frac{3\sqrt[]{5}}{2} \\ x_1=\frac{13}{2}+\frac{3\sqrt[]{5}}{2} \\ x_2=\frac{13}{2}-\frac{3\sqrt[]{5}}{2} \end{gathered}[/tex]
3) Hence the number that should be added to complete the square is 169/4 on both sides
