(a) find the vertex of the quadratic.

the quadratic function can be written as:

\(\displaystyle{y}={x}^{{2}}−{14}{x}+{5}\)

\(\displaystyle{y}={x}^{{2}}−{14}{x}+{7}^{{2}}−{7}^{{2}}+{5}\)

\(\displaystyle{y}={x}^{{2}}−{2}{\left({x}\right)}{\left({7}\right)}+{7}^{{2}}−{49}+{5}\)

\(\displaystyle{y}={\left({x}−{7}\right)}^{{2}}−{44}\) (1)

as we know that the quadratic function \(\displaystyle{y}={a}{x}^{{2}}+{b}{x}+{c}\) in standard form can be written in vertex form having vertex at (h,k) as:

\(\displaystyle{y}={a}{\left({x}−{h}\right)}^{{2}}+{k}\)

therefore by comparing (1) with the equation \(\displaystyle{y}={a}{\left({x}−{h}\right)}^{{2}}+{k}\), we get

a=1, h=7 and k=−44

therefore the vertex of the quadratic is (h,k)=(7,−44)

(b) write the quadratic in vertex form.

the equation (1) is the equation of the quadratic in vertex form.

therefore the equation of the given quadratic in vertex form is:

\(\displaystyle{y}={\left({x}−{7}\right)}^{{2}}−{44}\)

the quadratic function can be written as:

\(\displaystyle{y}={x}^{{2}}−{14}{x}+{5}\)

\(\displaystyle{y}={x}^{{2}}−{14}{x}+{7}^{{2}}−{7}^{{2}}+{5}\)

\(\displaystyle{y}={x}^{{2}}−{2}{\left({x}\right)}{\left({7}\right)}+{7}^{{2}}−{49}+{5}\)

\(\displaystyle{y}={\left({x}−{7}\right)}^{{2}}−{44}\) (1)

as we know that the quadratic function \(\displaystyle{y}={a}{x}^{{2}}+{b}{x}+{c}\) in standard form can be written in vertex form having vertex at (h,k) as:

\(\displaystyle{y}={a}{\left({x}−{h}\right)}^{{2}}+{k}\)

therefore by comparing (1) with the equation \(\displaystyle{y}={a}{\left({x}−{h}\right)}^{{2}}+{k}\), we get

a=1, h=7 and k=−44

therefore the vertex of the quadratic is (h,k)=(7,−44)

(b) write the quadratic in vertex form.

the equation (1) is the equation of the quadratic in vertex form.

therefore the equation of the given quadratic in vertex form is:

\(\displaystyle{y}={\left({x}−{7}\right)}^{{2}}−{44}\)