To complete the square for the quadratic equation \( x^2 + 2x + 16 = 0 \), follow these steps:
-
Start by isolating the quadratic and linear terms on one side of the equation:
\[ x^2 + 2x + 16 = 0 \implies x^2 + 2x = -16 \]
-
To complete the square for the expression \( x^2 + 2x \), we take the coefficient of \( x \), which is \( 2 \), divide it by \( 2 \), and then square the result:
\[ \left(\frac{2}{2}\right)^2 = 1^2 = 1 \]
-
Add this number (1) to both sides of the equation:
\[ x^2 + 2x + 1 = -16 + 1 \]
-
Simplifying gives:
\[ x^2 + 2x + 1 = -15 \]
-
Now, the left side can be factored:
\[ (x + 1)^2 = -15 \]
So, the number that needs to be added to complete the square is 1.