Complete the square to write \(k^2 - 8k - 9\) in vertex form.
\(k^2 - 8k - 9\) is in standard form, therefore
- a = \(\answer {1}\)
- b = \(\answer {-8}\)
- c = \(\answer {-9}\)
This makes \(\frac {b}{2} = \answer {-4}\), and its square is \(\left ( \frac {b}{2} \right )^2 = \answer {16}\), which is added and subtracted to the expression. That way we have only add \(0\) to the expression and not changed any values.
\[ k^2 - 8k + \answer {16} - \answer {16} + 11 \]
The “added” number is grouped together with the quadratic and linear terms to form a perfect square.
\[ \left ( k - \answer {4} \right )^2 - 16 + 11 \]
And, we can combine the constant terms.
\[ ( k - 4 )^2 - 15 \]