Questions: Add the proper constant to the binomial so that the resulting trinomial is a perfect square. Then factor the trinomial. [ z^2+14 z ] Type the missing term that completes the square. [ z^2+14 z+square ] (Simplify your answer. Type an integer or a fraction.)

Solution Steps

To complete the square for the binomial \( z^2 + 14z \), we need to add a constant term that makes it a perfect square trinomial. The constant term can be found by taking half of the coefficient of \( z \), squaring it, and adding it to the binomial.

The given binomial is \( z^2 + 14z \). The coefficient of \( z \) is \( b = 14 \).

To complete the square, we take half of the coefficient \( b \) and square it: \[ \left( \frac{b}{2} \right)^2 = \left( \frac{14}{2} \right)^2 = 7^2 = 49 \] Thus, the constant term that completes the square is \( 49 \).

Now, we can express the trinomial as: \[ z^2 + 14z + 49 \] This can be factored as: \[ (z + 7)^2 \]

Final Answer