Completing the square Fill in the blank with a number to make the expression a perfect square. \( y^{2}+16 y+\square \)

To complete the square for the expression \(y^{2}+16y+\square\), you need to take half of the coefficient of \(y\), which is \(16\), and square it. Half of \(16\) is \(8\), and squaring it gives you \(64\). So, you would fill in the blank with \(64\) to get the perfect square trinomial \(y^{2}+16y+64\), which can be factored as \((y+8)^{2}\). Completing the square is not just a math trick; it has practical applications in various fields such as physics and economics! For instance, when optimizing revenue or understanding projectile motion, completing the square helps to find maximum or minimum values of quadratic functions. It's like transforming a messy equation into a neat and tidy expression that tells you exactly where the peak or the trough lies!