\(\displaystyle 12x^{2}+13x \ = \ 4 \ \implies \ 12x^{2}+13x-4 \ = \ 0.\)
\(\displaystyle 12\bigg[x^{2}+\frac{13x}{12}-\frac{1}{3}\bigg] \ = \ 0\)
\(\displaystyle 12\bigg[\bigg(x+\frac{13}{24}\bigg)^{2}-\frac{361}{576}\bigg] \ = \ 0, \ this \ completes \ the \ square.\)
\(\displaystyle Check: \ 12\bigg[x^{2}+\frac{26x}{24}+\frac{169}{576}-\frac{361}{576}\bigg] \ = \ 0\)
\(\displaystyle = \ 12\bigg[x^{2}+\frac{13x}{12}-\frac{192}{576}\bigg] \ = \ 0\)
\(\displaystyle =12\bigg[x^{2}+\frac{13x}{12}-\frac{1}{3}\bigg] \ = \ 0\)
\(\displaystyle = \ 12x^{2}+13x-4 \ = \ 0\)
\(\displaystyle Your \ query: \ \frac{169}{576}+k \ = \ -\frac{1}{3}, \ k \ being \ the \ unknown \ we \ wish \ to \ know.\)
\(\displaystyle Hence, \ k \ = \ -\frac{1}{3}-\frac{169}{576} \ = \ -\frac{361}{576}, \ capishe.\)
