Solve for y over the real numbers:

\(\displaystyle{y}^{{{2}}}-{7}{y}-{5}={0}\)

Hint: Using the quadratic formula, solve for y.

\(\displaystyle{\left\lbrace{y}\right\rbrace}={\frac{{{\left\lbrace{\left\lbrace{7}\right\rbrace}\pm\sqrt{{{\left\lbrace{\left\lbrace{\left(-{\left\lbrace{7}\right\rbrace}\right)}\right\rbrace}^{{{2}}}-{\left\lbrace{4}\right\rbrace}{\left\lbrace{\left(-{\left\lbrace{5}\right\rbrace}\right)}\right\rbrace}\right\rbrace}}}\right\rbrace}}}{{{\left\lbrace{2}\right\rbrace}}}}={\frac{{{\left\lbrace{\left\lbrace{7}\right\rbrace}\pm\sqrt{{{\left\lbrace{\left\lbrace{49}\right\rbrace}+{\left\lbrace{20}\right\rbrace}\right\rbrace}}}\right\rbrace}}}{{{\left\lbrace{2}\right\rbrace}}}}={\frac{{{\left\lbrace{\left\lbrace{7}\right\rbrace}\pm\sqrt{{{\left\lbrace{\left\lbrace{69}\right\rbrace}\right\rbrace}}}\right\rbrace}}}{{{\left\lbrace{2}\right\rbrace}}}}\)

Answer:

\(\displaystyle{\left\lbrace{y}\right\rbrace}={\frac{{{\left\lbrace{\left\lbrace{7}\right\rbrace}+\sqrt{{{\left\lbrace{\left\lbrace{69}\right\rbrace}\right\rbrace}}}\right\rbrace}}}{{{\left\lbrace{2}\right\rbrace}}}}{\left\lbrace\quad\text{or}\quad\right\rbrace}{\left\lbrace{y}\right\rbrace}={\frac{{{\left\lbrace{\left\lbrace{7}\right\rbrace}-\sqrt{{{\left\lbrace{\left\lbrace{69}\right\rbrace}\right\rbrace}}}\right\rbrace}}}{{{\left\lbrace{2}\right\rbrace}}}}\)

\(\displaystyle{y}^{{{2}}}-{7}{y}-{5}={0}\)

Hint: Using the quadratic formula, solve for y.

\(\displaystyle{\left\lbrace{y}\right\rbrace}={\frac{{{\left\lbrace{\left\lbrace{7}\right\rbrace}\pm\sqrt{{{\left\lbrace{\left\lbrace{\left(-{\left\lbrace{7}\right\rbrace}\right)}\right\rbrace}^{{{2}}}-{\left\lbrace{4}\right\rbrace}{\left\lbrace{\left(-{\left\lbrace{5}\right\rbrace}\right)}\right\rbrace}\right\rbrace}}}\right\rbrace}}}{{{\left\lbrace{2}\right\rbrace}}}}={\frac{{{\left\lbrace{\left\lbrace{7}\right\rbrace}\pm\sqrt{{{\left\lbrace{\left\lbrace{49}\right\rbrace}+{\left\lbrace{20}\right\rbrace}\right\rbrace}}}\right\rbrace}}}{{{\left\lbrace{2}\right\rbrace}}}}={\frac{{{\left\lbrace{\left\lbrace{7}\right\rbrace}\pm\sqrt{{{\left\lbrace{\left\lbrace{69}\right\rbrace}\right\rbrace}}}\right\rbrace}}}{{{\left\lbrace{2}\right\rbrace}}}}\)

Answer:

\(\displaystyle{\left\lbrace{y}\right\rbrace}={\frac{{{\left\lbrace{\left\lbrace{7}\right\rbrace}+\sqrt{{{\left\lbrace{\left\lbrace{69}\right\rbrace}\right\rbrace}}}\right\rbrace}}}{{{\left\lbrace{2}\right\rbrace}}}}{\left\lbrace\quad\text{or}\quad\right\rbrace}{\left\lbrace{y}\right\rbrace}={\frac{{{\left\lbrace{\left\lbrace{7}\right\rbrace}-\sqrt{{{\left\lbrace{\left\lbrace{69}\right\rbrace}\right\rbrace}}}\right\rbrace}}}{{{\left\lbrace{2}\right\rbrace}}}}\)