To complete the trinomial \( k^2 - 7k \) so that it is a perfect square and then factor it, we can follow these steps:
1. Identify the coefficient of \( k \):
The coefficient of \( k \) is \(-7\).
2. Calculate the term needed to complete the square:
To complete the square, we take half of the coefficient of \( k \), and then square it.
So, \(\left(\frac{-7}{2}\right)^2 = \left(\frac{-7}{2}\right) \cdot \left(\frac{-7}{2}\right) = \frac{49}{4}\).
3. Add and subtract the calculated term inside the equation:
Insert \(\frac{49}{4}\) to complete the trinomial.
Therefore, the trinomial becomes:
[tex]\[
k^2 - 7k + \frac{49}{4}
\][/tex]
4. Factor the completed square trinomial:
For a trinomial \( k^2 + 2bk + b^2 \), it factors to \((k + b)^2\). Here, we need to rewrite our trinomial in that form.
Since \( b = \frac{-7}{2} \), the trinomial \( k^2 - 7k + \frac{49}{4} \) factors to:
[tex]\[
\left(k - \frac{7}{2}\right)^2
\][/tex]
So the final solutions are:
- The missing term that completes the square is \(\frac{49}{4}\).
- The trinomial factors to [tex]\(\left(k - \frac{7}{2}\right)^2\)[/tex].
