To express \(\displaystyle{x}^{{{2}}}+{8}{x}\) in the form (x+a)2+b, we need to complete the square.

For an expression of the form \(\displaystyle{x}^{{{2}}}+{B}{x}\), we can complete the square by adding and subtracting \(\displaystyle{\frac{{{B}}}{{{2}}}}^{{{2}}}\). You need to add and subtract this number so the new expression will be equivalent to the original expression.

For x2+8x, B=8 so we need to add and subtract (82)2=42=16. This gives:

\(\displaystyle{x}^{{{2}}}+{8}{x}={x}^{{{2}}}+{8}{x}+{16}−{16}\)

Group the first three terms to form a perfect square trinomial:

\(\displaystyle={\left({x}^{{2}}+{8}{x}+{16}\right)}−{16}\)

A perfect square trinomial of the form \(\displaystyle{x}^{{{2}}}+{B}{x}+{\frac{{{B}}}{{{2}}}}^{{{2}}}\) factors to \(\displaystyle{\left({x}+{B}{2}\right)}{2}{\left({x}+{2}{B}\right)}^{{{2}}}\).

Since \(\displaystyle{\frac{{{B}}}{{{2}}}}={\frac{{{8}}}{{{2}}}}={4},{t}{h}{e}{n}\ {x}^{{{2}}}+{8}{x}+{16}\) factors to \(\displaystyle{\left({x}+{4}\right)}^{{{2}}}\). Therefore:

\(\displaystyle{\left({x}^{{{2}}}+{8}{x}+{16}\right)}−{16}={\left({x}+{4}\right)}^{{{2}}}−{16}\)

For an expression of the form \(\displaystyle{x}^{{{2}}}+{B}{x}\), we can complete the square by adding and subtracting \(\displaystyle{\frac{{{B}}}{{{2}}}}^{{{2}}}\). You need to add and subtract this number so the new expression will be equivalent to the original expression.

For x2+8x, B=8 so we need to add and subtract (82)2=42=16. This gives:

\(\displaystyle{x}^{{{2}}}+{8}{x}={x}^{{{2}}}+{8}{x}+{16}−{16}\)

Group the first three terms to form a perfect square trinomial:

\(\displaystyle={\left({x}^{{2}}+{8}{x}+{16}\right)}−{16}\)

A perfect square trinomial of the form \(\displaystyle{x}^{{{2}}}+{B}{x}+{\frac{{{B}}}{{{2}}}}^{{{2}}}\) factors to \(\displaystyle{\left({x}+{B}{2}\right)}{2}{\left({x}+{2}{B}\right)}^{{{2}}}\).

Since \(\displaystyle{\frac{{{B}}}{{{2}}}}={\frac{{{8}}}{{{2}}}}={4},{t}{h}{e}{n}\ {x}^{{{2}}}+{8}{x}+{16}\) factors to \(\displaystyle{\left({x}+{4}\right)}^{{{2}}}\). Therefore:

\(\displaystyle{\left({x}^{{{2}}}+{8}{x}+{16}\right)}−{16}={\left({x}+{4}\right)}^{{{2}}}−{16}\)