To complete the square for the quadratic expression \( t^2 - 19t \), you'll want to find a number that makes this a perfect square trinomial.
The process involves taking half of the coefficient of \( t \), squaring it, and then adding it to the expression.
- Take the coefficient of \( t \), which is \(-19\).
- Half of \(-19\) is \(-\frac{19}{2}\).
- Squaring this gives: \[ \left(-\frac{19}{2}\right)^2 = \frac{361}{4}. \]
Now we add this value to \( t^2 - 19t \) to complete the square:
\[ t^2 - 19t + \frac{361}{4}. \]
Thus, the completed square form is:
\[ \left(t - \frac{19}{2}\right)^2. \]
So, the number that needs to be added to make the polynomial a perfect-square quadratic is \( \frac{361}{4} \).