Step 1

a) \(\displaystyle{6}{x}^{{{2}}}+{x}-{12}\)

Factorization:

\(\displaystyle{6}{x}{12}={72}\)

\(\displaystyle{72}={9}{x}{8}\)

\(\displaystyle{6}{x}^{{{2}}}+{9}{x}-{8}{x}-{12}={0}\)

\(\displaystyle{3}{x}{\left({2}{x}+{3}\right)}-{4}{\left({2}{x}+{3}\right)}={0}\)

\(\displaystyle{\left({2}{x}+{3}\right)}{\left({3}{x}-{4}\right)}={0}\)

\(\displaystyle{2}{x}+{3}={0}\) or \(\displaystyle{3}{x}-{4}={0}\)

\(\displaystyle{x}={\frac{{-{3}}}{{{2}}}}{x}={\frac{{{4}}}{{{3}}}}\)

Step 2

Method:

\(\displaystyle{x}={\frac{{-{b}\pm\sqrt{{{b}^{{{2}}}-{4}{a}{c}}}}}{{{2}{a}}}}\)

\(\displaystyle{x}={\frac{{-{1}\pm\sqrt{{{1}+{4}{x}{6}{x}{12}}}}}{{{2}{x}{6}}}}\)

\(\displaystyle{x}={x}={\frac{{-{1}\pm{17}}}{{{12}}}}\)

\(\displaystyle{x}={\frac{{{16}}}{{{12}}}}\) or \(\displaystyle{x}={\frac{{-{18}}}{{{12}}}}\)

\(\displaystyle{x}={\frac{{{4}}}{{{3}}}}\) or \(\displaystyle{x}={\frac{{-{3}}}{{{2}}}}\)

a) \(\displaystyle{6}{x}^{{{2}}}+{x}-{12}\)

Factorization:

\(\displaystyle{6}{x}{12}={72}\)

\(\displaystyle{72}={9}{x}{8}\)

\(\displaystyle{6}{x}^{{{2}}}+{9}{x}-{8}{x}-{12}={0}\)

\(\displaystyle{3}{x}{\left({2}{x}+{3}\right)}-{4}{\left({2}{x}+{3}\right)}={0}\)

\(\displaystyle{\left({2}{x}+{3}\right)}{\left({3}{x}-{4}\right)}={0}\)

\(\displaystyle{2}{x}+{3}={0}\) or \(\displaystyle{3}{x}-{4}={0}\)

\(\displaystyle{x}={\frac{{-{3}}}{{{2}}}}{x}={\frac{{{4}}}{{{3}}}}\)

Step 2

Method:

\(\displaystyle{x}={\frac{{-{b}\pm\sqrt{{{b}^{{{2}}}-{4}{a}{c}}}}}{{{2}{a}}}}\)

\(\displaystyle{x}={\frac{{-{1}\pm\sqrt{{{1}+{4}{x}{6}{x}{12}}}}}{{{2}{x}{6}}}}\)

\(\displaystyle{x}={x}={\frac{{-{1}\pm{17}}}{{{12}}}}\)

\(\displaystyle{x}={\frac{{{16}}}{{{12}}}}\) or \(\displaystyle{x}={\frac{{-{18}}}{{{12}}}}\)

\(\displaystyle{x}={\frac{{{4}}}{{{3}}}}\) or \(\displaystyle{x}={\frac{{-{3}}}{{{2}}}}\)